# Computing rational powers of monomial ideals

Pratik Dongre, Benjamin Drabkin, Josiah Lim, Ethan Partida, Ethan Roy, Dylan
Ruff, Alexandra Seceleanu, Tingting Tang  Indian Institute of Information
Technology, Nagpur<EMAIL_ADDRESS>Singapore University of
technology and Design<EMAIL_ADDRESS>Johns Hopkins University
<EMAIL_ADDRESS>Brown University<EMAIL_ADDRESS>The University of
Texas at Austin<EMAIL_ADDRESS>University of Toronto
<EMAIL_ADDRESS>University of Nebraska–Lincoln<EMAIL_ADDRESS>San Diego State University<EMAIL_ADDRESS>
###### Abstract.

This paper concerns fractional powers of monomial ideals. Rational powers of a
monomial ideal generalize the integral closure operation as well as recover
the family of symbolic powers. They also highlight many interesting
connections to the theory of convex polytopes. We provide multiple algorithms
for computing the rational powers of a monomial ideal. We also introduce a
mild generalization allowing real powers of monomial ideals. An important
result is that given any monomial ideal $I$, the function taking a real number
to the corresponding real power of $I$ is a step function which is left
continuous and has rational discontinuity points.

###### Key words and phrases:

monomial ideals, rational powers, Newton polyhedron, computational algebra,
jumping numbers

###### 2020 Mathematics Subject Classification:

Primary 13F55, 13F20.

##  1\. Introduction

An ideal of the polynomial ring $R=K[x_{1},\ldots,x_{d}]$ with coefficients in
a field $K$ is a monomial ideal if it is generated by monomials.

In this paper, we study a notion of powers for monomial ideals, where the
exponents are allowed to be real numbers as follows: for $r\in{\mathbb{R}}$,
$r>0$ we define the $r$-th real power of a monomial ideal, $I$, denoted
$\overline{I^{r}}$ to be the monomial ideal whose exponent set consists of
(integer) lattice points in the $r$-th dilate of the Newton polyhedron of $I$;
see 3.1. We emphasize that $\overline{I^{r}}$ is an ideal of the polynomial
ring $R$, and in particular the monomial generators of $\overline{I^{r}}$ have
natural number exponents. Thus our notion of real powers of ideals bears no
overlap with work taking place in a ring where monomials are allowed to have
real number exponents. Prominent examples of work in the latter context are
[ISW13, ASW15, Mil20].

Our notion of real powers is inspired by, and in fact coincides when
$r\in{\mathbb{Q}}$, with the notion of rational powers, which can be defined
for arbitrary ideals, and have appeared previously in the literature in [HS06,
§10.5], [Knu06], [Rus07], [Ciu20], [Ciu21], [Lew20]. In these works, rational
powers come up in contexts ranging from valuation theory to intersection
theory and have application to establishing the Golod property. In particular,
[Lew20, Corollary 3.4] establishes a strong connection between rational powers
and the widely studied family of symbolic powers of monomial ideals. The above
mentioned applications have motivated and inspired us to seek effective
methods for handling rational powers from a computational standpoint.

The focus of this paper is twofold. First, we handle the task of computing
real powers of monomial ideals. One main result in this direction is 3.5,
where we show that the generators of a specified real power of a monomial
ideal can be confined within a bounded convex region depending only on the
exponent and the Newton polytope of the ideal. We complement this theoretical
insight with a series of algorithms, Algorithm 1, Algorithm 2, Algorithm 3,
and Algorithm 4 which exploit different features of the problem to provide
practical solutions for computing real powers of monomial ideals.

Our second aim is to study continuity properties of the exponentiation
function where the base is a monomial ideal. Being able to do this provides
motivation for working with real powers as opposed to the more common rational
powers. We find that the exponentiation function is a step function with
rational discontinuity points which we term jumping points. This leads to the
conclusion that all distinct real powers of a fixed monomial ideal are given
by rational exponents. Our main results on properties for the real
exponentiation function of a monomial ideal are contained in 5.2 (existence of
right limits) 5.6 (left continuity), 5.7 (step function), and 5.9 (jumping
numbers).

Our paper is organized as follows. After introducing the notions of Newton
polyhedron and integral closure in section 2, we turn our attention to real
powers of monomial ideals in section 3 and present algorithms capable of
computing these ideals in section 4. We end with studying continuity
properties and jumping numbers for exponentiation in section 5.

##  2\. Background on integral closure and the Newton polyhedron

Let ${\mathbb{R}}$ and ${\mathbb{R}}_{+}$ denote the real numbers and non-
negative real numbers respectively. We denote by ${\mathbb{N}}$ the set of non
negative integers.

Let $R=K[x_{1},\cdots,x_{d}]$ be a polynomial ring with coefficients in a
field $K$. Every monomial ideal $I$ in $R$ has a unique minimal monomial
generating set denoted $G(I)$. This is a set of monomials that generates $I$
and such that no element of $G(I)$ divides another element of $G(I)$. It is
customary to denote monomials in $R$ by the shorthand notation
${\mathbf{x}}^{\mathbf{a}}:=x_{1}^{a_{1}}\cdots x_{d}^{a_{d}}$, where
${\mathbf{a}}\in{\mathbb{N}}^{d}$. The bijective correspondence between
monomials ${\mathbf{x}}^{\mathbf{a}}$ and lattice points
${\mathbf{a}}\in{\mathbb{N}}^{n}$ gives rise to convex geometric
representations for monomial ideals, chief among which is the Newton
polyhedron.

######  Definition 2.1.

For any monomial ideal $I$ denote by ${\mathcal{L}}(I)$ the set of exponent
vectors of all monomials in $I$

${\mathcal{L}}(I)=\\{{\mathbf{a}}\mid{\mathbf{x}}^{\mathbf{a}}\in I\\}.$

The Newton polyhedron of $I$, denoted $NP(I)$, is the convex hull of
${\mathcal{L}}(I)$ in ${\mathbb{R}}^{d}$

$NP(I)=\operatorname{convex\ hull}{\mathcal{L}}(I)=\operatorname{convex\
hull}(\\{{\mathbf{a}}\mid{\mathbf{x}}^{\mathbf{a}}\in I\\}).$

The Newton polytope of $I$, denoted $\operatorname{np}(I)$, is the convex hull
of the exponent vectors of a minimal monomial generating set for $I$.

$\operatorname{np}(I)=\operatorname{convex\
hull}(\\{{\mathbf{a}}\mid{\mathbf{x}}^{\mathbf{a}}\in G(I)\\}).$

Notice that Newton polyhedra are unbounded, while Newton polytopes are bounded
convex bodies. Both are lattice polyhedra, meaning that their vertices have
integer coordinates. Their relationship can be described using the notion of
Minkowski sum.

######  Definition 2.2.

The Minkowski sum of subsets $A,B\subseteq{\mathbb{R}}^{n}$ is

$A+B=\\{{\mathbf{a}}+{\mathbf{b}}\mid{\mathbf{a}}\in A,{\mathbf{b}}\in B\\}.$

We also write $A-B=\\{{\mathbf{a}}-{\mathbf{b}}\mid{\mathbf{a}}\in
A,{\mathbf{b}}\in B\\}$.

The precise relationship between the Newton polyhedron and the Newton polytope
of $I$, established for example in [CEHH17, Lemma 5.2], is given by the
Minkowski sum decomposition

(2.1) $NP(I)=\operatorname{np}(I)+{\mathbb{R}}^{d}_{+},$

where ${\mathbb{R}}^{d}_{+}=\\{(a_{1},\ldots,a_{d})\in{\mathbb{R}}^{d}\mid
a_{i}\geq 0\\}$ denotes the positive orthant in ${\mathbb{R}}^{d}$.

By the version of Carathéodory’s theorem in [CEHH17, Theorem 5.2], any point
${\mathbf{a}}\in NP(I)$ is written as

(2.2)
${\mathbf{a}}=\lambda_{1}{\mathbf{t}}_{1}+\cdots+\lambda_{d}{\mathbf{t}}_{d}+c_{1}{\mathbf{e}}_{1}+\cdots+c_{d}{\mathbf{e}}_{d},$

with $\lambda_{i},c_{j}\geq 0,\sum_{i=1}^{d}\lambda_{i}=1$,
${\mathbf{t}}_{1},\ldots,{\mathbf{t}}_{d}\in\operatorname{np}(I)$, and
${\mathbf{e}}_{1},\ldots,{\mathbf{e}}_{d}$ standard basis vectors in
${\mathbb{R}}^{d}$. Thus one can reformulate equation (2.1) using
coordinatewise inequalities as

(2.3)
$NP(I)=\\{{\mathbf{a}}\in{\mathbb{R}}^{d}\mid{\mathbf{a}}\geq{\mathbf{b}}\text{
for some }{\mathbf{b}}\in\operatorname{np}(I)\\}$

While the containment ${\mathcal{L}}(I)\subseteq NP(I)\cap{\mathbb{N}}^{d}$
holds by definition, in general the sets of lattice points ${\mathcal{L}}(I)$
and $NP(I)\cap{\mathbb{N}}^{d}$ need not be equal. We recall below that the
set of lattice points in $NP(I)$ is in fact given by
$NP(I)\cap{\mathbb{N}}^{d}={\mathcal{L}}(\overline{I})$, where $\overline{I}$
is the integral closure of $I$.

######  Definition 2.3.

The integral closure of an ideal $I$ of a ring $R$ is the set of elements
$y\in R$ that satisfy an equation of integral dependence of the form

$y^{n}+m_{1}y^{n-1}+\cdots+m_{n-1}y+m_{n}=0\text{ where }m_{i}\in I^{i},n\geq
1.$

The integral closure of $I$ is denoted $\overline{I}$.

######  Remark 2.4.

It is shown in [HS06] that the description is significantly simpler if $I$ is
a monomial ideal. In this case one can give an alternate definition for the
integral closure

(2.4)
$\overline{I}=\left(\\{{\mathbf{x}}^{\mathbf{a}}\mid{\mathbf{x}}^{n{\mathbf{a}}}\in
I^{n}\text{ for some }n\in{\mathbb{N}}\\}\right).$

We recall below how the integral closure of a monomial ideal $I$ can be
described in terms of its Newton polyhedron. We also show that the minimal
generators of $\overline{I}$ lie at bounded lattice distance from the Newton
polytope $\operatorname{np}(I)$. In the following we use the notion of lattice
(or taxicab) distance between points in
${\mathbf{a}},{\mathbf{b}}\in{\mathbb{R}}^{d}$ defined as
$\operatorname{dist}({\mathbf{a}},{\mathbf{b}})=\sum_{i=1}^{d}|a_{i}-b_{i}|$.

######  Lemma 2.5.

Let $I$ be a monomial ideal in $K[x_{1},\ldots,x_{d}]$. Then

  1. (1)

$NP(I)\cap{\mathbb{N}}^{d}={\mathcal{L}}(\overline{I})$,

  2. (2)

$NP(\overline{I})=NP(I)$,

  3. (3)

(compare [HS06, Proposition 1.4.9]) if ${\mathbf{x}}^{\mathbf{a}}\in
G(\overline{I})$, then there exists ${\mathbf{b}}\in\operatorname{np}(I)$ such
that ${\mathbf{a}}\geq{\mathbf{b}}$ and

$\sum_{i=1}^{d}(a_{i}-b_{i})\leq d-1.$

###### Proof.

Statement (1) is well-known; see for example [HS06, Proposition 1.4.6].

(2) follows from (1) by noticing that, since $NP(I)$ is a lattice polyhedron
we have

$NP(I)=\operatorname{convex\
hull}(NP(I)\cap{\mathbb{N}}^{d})=\operatorname{convex\
hull}({\mathcal{L}}(\overline{I}))=NP(\overline{I}).$

(3) If ${\mathbf{a}}\in\operatorname{np}(I)$, the choice
${\mathbf{b}}={\mathbf{a}}$ works as claimed. We may thus assume
${\mathbf{a}}\not\in\operatorname{np}(I)$. By (2.3) there is
${\mathbf{y}}\in\operatorname{np}(I)$ such that the inequality
${\mathbf{a}}\geq{\mathbf{y}}$ is satisfied coordinatewise. Since
${\mathbf{a}}\in{\mathbb{N}}^{d}$, we have that
${\mathbf{a}}\geq\lceil{\mathbf{y}}\rceil:=(\lceil y_{1}\rceil,\ldots,\lceil
y_{d}\rceil)$ and since $\lceil{\mathbf{y}}\rceil\geq{\mathbf{y}}$, we have
$\lceil{\mathbf{y}}\rceil\in NP(I)$. As ${\mathbf{x}}^{\mathbf{a}}$ is a
minimal generator of $\overline{I}$, it follows that
${\mathbf{a}}=\lceil{\mathbf{y}}\rceil$.

Denote the unit hypercube in ${\mathbb{R}}^{d}$ by $H_{d}$; it has vertices
$\sum_{i\in S\subseteq[d]}{\mathbf{e}}_{i}$. Since $x^{\mathbf{a}}$ is a
minimal generator of $\overline{I}$, it follows that the only vertex of
${\mathbf{a}}-H_{d}$ that is in $NP(I)$ is ${\mathbf{a}}$. Moreover, since the
only lattice points in ${\mathbf{a}}-H_{d}$ are its vertices, the only lattice
point in $({\mathbf{a}}-H_{d})\cap NP(I)$ is ${\mathbf{a}}$. Finally, we have
${\mathbf{y}}\in{\mathbf{a}}-H_{d}$ because
${\mathbf{a}}=\lceil{\mathbf{y}}\rceil$.

Let ${\mathbf{z}}\in{\mathbb{N}}^{d}$ be any vertex of $\operatorname{np}(I)$.
From the previous considerations, we have
${\mathbf{z}}\not\in{\mathbf{a}}-H_{d}$. Since $\operatorname{np}(I)$ is
convex, the line segment $[{\mathbf{y}},{\mathbf{z}}]$ is contained in
$\operatorname{np}(I)$. Let ${\mathbf{b}}$ be the intersection point of this
line segment with the boundary of the polytope ${\mathbf{a}}-H_{d}$. Such an
intersection point exists since ${\mathbf{y}}$ is inside and ${\mathbf{z}}$ is
outside ${\mathbf{a}}-H_{d}$. Then ${\mathbf{b}}\in\operatorname{np}(I)$ and
$\lceil{\mathbf{b}}\rceil$ is a vertex of ${\mathbf{a}}-H_{d}$ that belongs to
$NP(I)$; thus we have $\lceil{\mathbf{b}}\rceil={\mathbf{a}}$. Furthermore,
since ${\mathbf{b}}\neq{\mathbf{a}}-\mathbf{1}$, and ${\mathbf{b}}$ is on the
boundary of ${\mathbf{a}}-H_{d}$, it follows that for some $1\leq i\leq d$ we
have $b_{i}=a_{i}$. Hence we obtain $\sum_{i=1}^{d}(a_{i}-b_{i})\leq d-1$, as
claimed.

∎

##  3\. Real powers of monomial ideals

We now discuss powers of monomial ideals with real exponents, termed real
powers, and their relationship to integral closure.

######  Definition 3.1.

Fix a real number $r\geq 0$. We define the $r$-th real power of a monomial
ideal, $I$, to be

$\overline{I^{r}}=\left(\\{{\mathbf{x}}^{\mathbf{a}}\mid{\mathbf{a}}\in r\cdot
NP(I)\cap{\mathbb{N}}^{d}\\}\right).$

When $r\in{\mathbb{Q}}$ we will refer to $\overline{I^{r}}$ as the $r$-th
rational power of $I$.

Rational powers of monomial ideals have appeared previously in the literature
under the following definition and notation, see [HS06, Definition 10.5.1]:
the $r$-th rational power of an arbitrary ideal $I$ of a ring $R$ for
$r=\frac{p}{q}$ with $p,q\in{\mathbb{N}},q\neq 0$ is the ideal

(3.1) $I_{r}:=\\{y\in R\mid y^{q}\in\overline{I^{p}}\\},$

where $\overline{I^{p}}$ denotes the integral closure of the $p$-th ordinary
power of $I$, $I^{p}$. In the following we show that these two definitions
agree, i.e., $I_{r}=\overline{I^{r}}$ whenever $r\in{\mathbb{Q}}$ and
furthermore for natural exponents $r\in{\mathbb{N}}$ the $r$-th real power
agrees with the integral closure of the $r$-th ordinary power of $I$, $I^{r}$.

Our notation for real powers deviates from that in (3.1), which is more
established in the literature, in favor of being intentionally consistent with
the notation for integral closure, since these notions agree for
$r\in{\mathbb{N}}$ as shown in the following lemma.

######  Lemma 3.2.

Let $I$ be a monomial ideal. Then

  1. (1)

If $r\in{\mathbb{N}}$, then the $r$-th real power of $I$ is equal to the
integral closure of the $r$-th ordinary power $I^{r}$. In particular, the
first rational power of $I$, $\overline{I^{1}}$, is the integral closure of
$I$. Moreover, the $r$-th real power of $I$ is integrally closed.

  2. (2)

If $r\in{\mathbb{Q}}$ then the $r$-th real power of $I$ in 3.1 agrees with the
$r$-th rational power of $I$, $I_{r}$, in (3.1).

###### Proof.

(1) By definition, a monomial $\mathbf{x}^{\mathbf{a}}$ is an element of the
$r$-th real power of $I$ if and only if $\mathbf{a}\in r\cdot NP(I)$. Noting
that $r\cdot NP(I)=NP(I^{r})$ if $r\in{\mathbb{N}}$, the latter condition is
equivalent to $\mathbf{a}\in NP(I^{r})$. Now by 2.5 (1), we have
$\mathbf{a}\in NP(I^{r})\cap{\mathbb{N}}^{d}$ if and only if
$\mathbf{x}^{\mathbf{a}}$ is an element of the integral closure of $I^{r}$ if
and only if $\mathbf{x}^{\mathbf{a}}$ is an element of the integral closure of
$\overline{I^{r}}$.

(2) Let $r=\frac{p}{q}$ with $p,q\in{\mathbb{N}},q\neq 0$ and let
${\mathbf{x}}^{\mathbf{a}}$ be a monomial. By (3.1),
${\mathbf{x}}^{\mathbf{a}}\in I_{r}$ holds if and only if we have
${\mathbf{x}}^{q{\mathbf{a}}}\in\overline{I^{p}}$, equivalently
$q{\mathbf{a}}\in NP(\overline{I^{p}})=NP(I^{p})=pNP(I)$. In turn, the last
assertion is equivalent to ${\mathbf{a}}\in rNP(I)\cap{\mathbb{N}}^{d}$ and by
3.1 this holds if and only if ${\mathbf{x}}^{\mathbf{a}}\in\overline{I^{r}}$.
∎

Using 2.5, for $r\in{\mathbb{Q}}_{+}$ we aim to confine the minimal generators
of $\overline{I^{r}}$ to a bounded convex set, which will be obtained by
Minkowski sum. In order to define this convex set we introduce the unit
simplex in $d$-dimensional space,

$S_{d}=\\{{\mathbf{a}}=(a_{1},\ldots,a_{d})\in{\mathbb{R}}^{d}\mid
a_{1}+\cdots+a_{d}\leq 1,a_{i}\geq 0\text{ for }1\leq i\leq d\\}.$

In the metric space ${\mathbb{R}}^{d}$ endowed with the lattice distance, the
unit simplex is the non negative portion of the ball of radius one centered at
the origin. Denoting the origin in ${\mathbb{R}}^{d}$ by $\bf 0$, this
observation yields an alternate description

$S_{d}=\\{{\mathbf{a}}\in{\mathbb{R}}^{d}\mid{\mathbf{a}}\geq\bf 0,\
\operatorname{dist}({\mathbf{a}},\bf 0)\leq 1\\}.$

######  Remark 3.3.

2.5 (3) can be reformulated using this notation as follows: If $I$ is a
monomial ideal and ${\mathbf{x}}^{\mathbf{a}}\in G(\overline{I})$, then
${\mathbf{a}}\in\operatorname{np}(I)+(d-1)\cdot S_{d}$.

The following technical result shall prove very useful for our purposes.

######  Lemma 3.4.

Let $\mathbf{x}^{\mathbf{a}}$ be a minimal generator of $\overline{I^{r}}$,
where $r=\frac{p}{q}$ is a positive rational number. Then there exists a
minimal generator $\mathbf{x}^{\mathbf{b}}$ of $\overline{I^{p}}$ such that
$q{\mathbf{a}}-{\mathbf{b}}\in d(q-1)\cdot S_{d}$.

###### Proof.

By 3.2 (2), we obtain ${\mathbf{x}}^{q{\mathbf{a}}}\in\overline{I^{p}}$. Thus
there exists a minimal generator ${\mathbf{x}}^{\mathbf{b}}\in
G(\overline{I^{p}})$ such that ${\mathbf{x}}^{\mathbf{b}}$ divides
${\mathbf{x}}^{q{\mathbf{a}}}$. This implies ${\mathbf{b}}\leq q{\mathbf{a}}$,
that is, $b_{i}\leq qa_{i}$ for all $1\leq i\leq d$. Suppose that
$q{\mathbf{a}}-{\mathbf{b}}\not\in d(q-1)\cdot S_{d}$. Then the inequality

$\sum_{i=1}^{d}(qa_{i}-b_{i})\geq d(q-1)+1$

follows by integrality. Applying the pigeon-hole principle, we find that there
must exist $i_{0}\in\\{1,\ldots,d\\}$ such that $qa_{i_{0}}-b_{i_{0}}\geq q$.
Rewriting, we get that $q(a_{i_{0}}-1)\geq b_{i_{0}}$. We can now set
${\mathbf{a}}^{\prime}={\mathbf{a}}-{\mathbf{e}}_{i_{0}}$ and with this
notation we find

${\mathbf{b}}\leq
q(a_{1},\ldots,a_{i_{0-1}},a_{i_{0}}-1,a_{i_{0+1}},\ldots,a_{d})=q{\mathbf{a}}^{\prime}.$

Thus ${\mathbf{x}}^{\mathbf{b}}$ divides
${\mathbf{x}}^{q{\mathbf{a}}^{\prime}}$ and
${\mathbf{x}}^{q{\mathbf{a}}^{\prime}}$ is an element of $\overline{I^{p}}$.
Applying 3.2 (2) again, this yields that,
${\mathbf{x}}^{{\mathbf{a}}^{\prime}}\in\overline{I^{r}}$, which contradicts
that ${\mathbf{x}}^{\mathbf{a}}$ is a minimal generator of $\overline{I^{r}}$.
∎

We are now able to describe a bounded convex set which contains the minimal
generators of a rational power for a monomial ideal. The following result
constitutes the basis for our Minkowski algorithm described in Algorithm 1.
See also 4.2 for an illustration of the convex set $\mathcal{C}(I,r)$ defined
below.

######  Theorem 3.5.

Let $I$ be a monomial ideal in $K[x_{1},\ldots,x_{d}]$. If $r=\frac{p}{q}$ is
a positive rational number and ${\mathbf{x}}^{\mathbf{a}}\in
G(\overline{I^{r}})$, then ${\mathbf{a}}$ is in the following bounded convex
set

(3.2)
$\mathcal{C}(I,r)=r\cdot\operatorname{np}(I)+\left(d-\frac{1}{q}\right)\cdot
S_{d}.$

Moreover, if ${\mathbf{a}}\in\mathcal{C}(I,r)\cap{\mathbb{N}}^{d}$, then
${\mathbf{x}}^{\mathbf{a}}\in\overline{I^{r}}$ and thus
$\overline{I^{r}}=(\\{{\mathbf{x}}^{\mathbf{a}}\mid{\mathbf{a}}\in\mathcal{C}(I,r)\cap{\mathbb{N}}^{d})\\}$.

###### Proof.

By 3.4, there exists a minimal generator of $\overline{I^{p}}$,
$\mathbf{x}^{\mathbf{b}}$, such that

$q{\mathbf{a}}-{\mathbf{b}}\in d(q-1)\cdot S_{d}$

and from 3.3 applied to the monomial ideal $I^{p}$ we have that

${\mathbf{b}}\in\operatorname{np}(I^{p})+\left(d-1\right)\cdot
S_{d}=p\cdot\operatorname{np}(I)+\left(d-1\right)\cdot S_{d}.$

Combining the displayed statements, we obtain

$\displaystyle q{\mathbf{a}}$ $\displaystyle\in
p\cdot\operatorname{np}(I)+(d-1)\cdot S_{d}+d(q-1)\cdot S_{d}$
$\displaystyle\iff{\mathbf{a}}$
$\displaystyle\in\frac{p}{q}\cdot\operatorname{np}(I)+\frac{d-1}{q}\cdot
S_{d}+\frac{d(q-1)}{q}\cdot S_{d}$ $\displaystyle\iff{\mathbf{a}}$
$\displaystyle\in r\cdot\operatorname{np}(I)+\left(d-\frac{1}{q}\right)\cdot
S_{d}.$

Finally, since $S_{d}\subseteq{\mathbb{R}}_{+}^{d}$, we have that
$\mathcal{C}(I,r)\subseteq r\cdot NP(I)$ by (2.1). Thus if
${\mathbf{a}}\in\mathcal{C}(I,r)\cap{\mathbb{N}}^{d}$, then ${\mathbf{a}}\in
r\cdot NP(I)$ which yields ${\mathbf{x}}^{\mathbf{a}}\in\overline{I^{r}}$
according to 3.1. The identity
$\overline{I^{r}}=({\mathbf{x}}^{\mathbf{a}}\mid{\mathbf{a}}\in\mathcal{C}(I,r)\cap{\mathbb{N}}^{d})$
follows from the previous assertions. ∎

######  Remark 3.6.

While the previous theorem does not require the rational number
$r=\frac{p}{q}$ to have $\gcd(p,q)=1$, in applications is desirable to work
with the reduced form of $r$ in order to obtain the smallest possible region
$\mathcal{C}(I,r)$.

##  4\. Algorithms for computing real powers

Several algorithms are proposed below for computing real powers of monomial
ideals.

Our algorithms rely on several auxiliary computational tasks, which are highly
non trivial, but can be performed currently by computer algebra systems such
as [GS] or [4ti2]. Specifically, we assume that independent routines are used
to compute the Newton polyhedron or polytope for a given monomial ideal. For
this reason, we take these convex bodies as input for our algorithms. For
Algorithm 1 we additionally assume the existence of a routine that finds all
the lattice points in a bounded convex polytope. This task is discussed in
detail in [DLHTY04].

###  4.1. Minkowski Algorithm

Our first algorithm uses the ideas presented in 3.5 and illustrated in 4.2 to
confine the generators of a real power $\overline{I^{r}}$ within a convex
region of bounded lattice distance from the Newton polytope
$\operatorname{np}(I)$.

Input: the Newton polytope $\operatorname{np}(I)$ of an ideal $I$, a rational
number $r=\frac{p}{q}\in{\mathbb{Q}}_{+}$

Output: a list of monomial generators for the ideal $\overline{I^{r}}$

1

/* Scaled newton polytope of I */

2

3scalednp := $r\cdot np(I)$

/* Bounded convex set, as given by 3.5 */

4

5$d:=$ dimension of the polynomial ring containing $I$

6simplex := $d$-dimensional simplex with vertices at
$\\{\mathbf{0},(d-\frac{1}{q})\mathbf{e}_{1},\ldots,(d-\frac{1}{q})\mathbf{e}_{d}\\}$.

7$C$ := minkowskiSum(scalednp, simplex)

/* Find all lattice points and their monomial counterpart */

8

9exponentVectors := latticePoints($C$)

10Initialize generators := $\emptyset$

11for ${\mathbf{b}}$ in exponentVectors do

12 generators := append(${\mathbf{x}}^{\mathbf{b}}$, generators)

/* Return the possibly non minimal monomial generators */

13

Return generators.

Algorithm 1 Minkowski Sum algorithm

######  Proposition 4.1.

If $I$ is a monomial ideal of a $d$-dimensional polynomial ring and
$r\in{\mathbb{R}}_{+}$, then Algorithm 1 returns a not necessarily minimal set
of monomial generators for $\overline{I^{r}}$.

###### Proof.

This follows from the assertion
$\overline{I^{r}}=(\\{{\mathbf{x}}^{\mathbf{a}}\mid{\mathbf{a}}\in\mathcal{C}(I,r)\\})\cap{\mathbb{N}}^{d})$
of 3.5. In Algorithm 1 the set $\mathcal{C}(I,r)$, termed $C$, is constructed
according to equation (3.2). ∎

######  Example 4.2.

Consider the ideal $I=(xy^{5},x^{2}y^{2},x^{4}y)$ and the rational number
$r=\frac{4}{3}$. Then one can determine that

$\overline{I^{4/3}}=(x^{2}y^{5},x^{2}y^{6},x^{2}y^{7},x^{3}y^{3},x^{3}y^{4},x^{3}y^{5},x^{3}y^{6},x^{4}y^{2},x^{4}y^{3},\newline
x^{4}y^{4},x^{4}y^{5},x^{5}y^{2},x^{5}y^{3},x^{6}y^{2})$

based on identifying the lattice points in the convex region

$\mathcal{C}\left(I,\frac{4}{3}\right)=\frac{4}{3}\cdot\operatorname{np}(I)+\frac{5}{3}\cdot
S_{2}$

given by 3.5. Note that $\overline{I^{4/3}}$ is minimally generated by
$G(\overline{I^{4/3}})=\\{x^{2}y^{5},x^{3}y^{3},x^{4}y^{2}\\}$. Thus,
Algorithm 1 does not in general identify the minimal generators, but rather a
possibly non minimal set of generators for $\overline{I^{r}}$. In the Figure
1, the region $\mathcal{C}(I,\frac{4}{3})$ is shaded in darker blue, while the
rest of the scaled polyhedron $\frac{4}{3}\cdot NP(I)$ is shaded in lighter
blue.

$0$$1$$2$$3$$4$$5$$6$$7$$8$$0$$1$$2$$3$$4$$5$$6$$7$$8$$9$$x$$y$$0$$1$$2$$3$$4$$5$$6$$7$$8$$0$$1$$2$$3$$4$$5$$6$$7$$8$$9$$x$$y$$r\cdot
NP(I)$Convex RegionMinimal GeneratorsInterior Points

Figure 1. Computing $\overline{(xy^{5},x^{2}y^{2},x^{4}y)^{4/3}}$ using the
Minkowski algorithm.

###  4.2. Hyperrectangle Algorithm

The next algorithms depend on the notion of the hyperrectangle of a scaled
Newton polyhedron, which is defined below.

######  Definition 4.3.

Given a monomial ideal $I$ of a $d$-dimensional polynomial ring and
$r\in{\mathbb{R}}_{+}$, define the set of scaled vertices of $I$ with respect
to $r$ to be

$\mathcal{V}(I,r)=\left\\{\lceil r\rceil:=\left(\lceil
ra_{1}\rceil,\dots,\lceil ra_{d}\rceil\right)\,|\,x^{\in}G(I)\right\\}.$

Let $\alpha=(\alpha_{1},\dots,\alpha_{d})\in\mathcal{V}(I,r)$. Define

(4.1)
$\min(\mathcal{V}(I,r),i)=\min\limits_{\alpha\in\mathcal{V}(I,r)}\alpha_{i}\quad\text{
and
}\quad\max(\mathcal{V}(I,r),i)=\max\limits_{\alpha\in\mathcal{V}(I,r)}\alpha_{i}.$

Finally, set the _hyperrectangle of $r\cdot NP(I)$_ to be the following set

$\displaystyle\operatorname{hype}(I,r)$
$\displaystyle=\\{{\mathbf{c}}=(c_{1},\dots,c_{d})\,|\,c_{i}\in[\min(\mathcal{V}(I,r),i),\,\max(\mathcal{V}(I,r),i)]\\}$
$\displaystyle=\prod_{i=1}^{d}[\min(\mathcal{V}(I,r),i),\,\max(\mathcal{V}(I,r),i)].$

We now see that the generators for the $r$-th real power of $I$ are among the
set of lattice points in $\operatorname{hype}(I,r)$.

######  Lemma 4.4.

Let $I$ be a monomial ideal and let $r\in{\mathbb{R}}_{+}$. Denote the set of
lattice points in $\operatorname{hype}(I,r)$ by $\mathcal{S}(I,r)$. Then

  1. (1)

$\lceil r\cdot\operatorname{np}(I)\rceil:=\\{(\lceil p_{1}\rceil,\ldots,\lceil
p_{d}\rceil)\mid{\mathbf{p}}\in
r\cdot\operatorname{np}(I)\\}\subseteq\mathcal{S}(I,r)$

  2. (2)

$\overline{I^{r}}$ is generated by a subset of the lattice points in
$\operatorname{hype}(I,r)$, more precisely

$\overline{I^{r}}=(\\{{\mathbf{x}}^{\mid}\in r\cdot
NP(I)\cap\operatorname{hype}(I,r)\cap{\mathbb{N}}^{d}\\}).$

###### Proof.

(1) Every point in ${\mathbf{p}}\in r\cdot\operatorname{np}(I)$ is a convex
combination of the vertices of this polytope, which are in the set
$V=\\{r\mid{\mathbf{x}}^{\in}G(I)\\}$. Since every coordinate $p_{i}$ of
${\mathbf{p}}$ is a convex combination of $i$-th coordinates of elements in
$V$ we obtain that $p_{i}\in[\min_{\in V}{a_{i}},\max_{\in V}{a_{i}}]$ for
$1\leq i\leq d$. Thus $\lceil
p_{i}\rceil\in[\min(\mathcal{V},i),\,\max(\mathcal{V},i)]$, which settles the
claim.

(2) Temporarily denote $J:=({\mathbf{x}}^{\mid}\in r\cdot
NP(I)\cap\mathcal{S}(I,r))$. Then $J\subseteq\overline{I^{r}}$ follows from
3.1. Now let $\in{\mathbb{N}}^{d}$ be such that
${\mathbf{x}}^{\in}\overline{I^{r}}$ and thus $\in r\cdot
NP(I)\cap{\mathbb{N}}^{d}$. From (2.1) we know

$r\cdot
NP(I)=r\cdot\operatorname{np}(I)+r\cdot{\mathbb{R}}_{+}^{d}=r\cdot\operatorname{np}(I)+{\mathbb{R}}_{+}^{d},$

thus there exists $\in r\cdot\operatorname{np}(I)$ such that $\geq$. Since
$\in{\mathbb{N}}^{d}$ it follows that $\geq\lceil\rceil=(\lceil
b_{1}\rceil,\ldots,\lceil b_{d}\rceil]$, where $\lceil\rceil\in\lceil
r\cdot\operatorname{np}(I)\rceil$. From part (1) it follows that
$\lceil\rceil\in\mathcal{S}(I,r)$ and from $\lceil\rceil\geq$ we deduce
$\lceil\rceil\in r\cdot\operatorname{np}(I)$ hence $\lceil\rceil\in r\cdot
NP(I)+{\mathbb{R}}_{+}^{d}$. We have thus shown that $\lceil\rceil\in r\cdot
NP(I)\cap\mathcal{S}(I,r)$, hence ${\mathbf{x}}^{\lceil\rceil}\in J$ and since
$\geq\lceil\rceil$ we deduce ${\mathbf{x}}^{\in}J$. Thus the containment
$\overline{I^{r}}\subseteq J$ has been established. ∎

Based on the previous result we produce the following algorithm.

Input: generators $G(I)$ and the Newton polyhedron $NP(I)$ of an ideal $I$, a
real number $r\in{\mathbb{R}}_{+}$

Output: a list of monomial generators for the ideal $\overline{I^{r}}$

1

2$d:=$ dimension of the polynomial ring containing $I$

3candidates := $\operatorname{hype}(I,r)\cap{\mathbb{N}}^{d}$

4Initialize generators := $\emptyset$

5for in candidates do

6 if in $r\cdot NP(I)$ then

7 generators := append(${\mathbf{x}}$, generators)

8

9

10Return generators.

Algorithm 2 Hyperrectangle algorithm

######  Proposition 4.5.

If $I$ is a monomial ideal and $r\in{\mathbb{R}}_{+}$, then Algorithm 2
returns a not necessarily minimal set of monomial generators for
$\overline{I^{r}}$.

###### Proof.

This follows from part (2) of 4.4. ∎

######  Example 4.6.

Figure 2 illustrates the set of lattice points in the hyperrectangle
$\operatorname{hype}(I,\frac{4}{3})$ for the ideal
$I=(xy^{5},x^{2}y^{2},x^{4}y)$. These are marked in solid yellow, solid purple
and hollow black. The set of generators returned by Algorithm 2 corresponds to
the yellow and purple lattice points, while the minimal generators correspond
to the purple points.

$0$$1$$2$$3$$4$$5$$6$$7$$8$$0$$1$$2$$3$$4$$5$$6$$7$$8$$9$$x$$y$$0$$1$$2$$3$$4$$5$$6$$7$$8$$0$$1$$2$$3$$4$$5$$6$$7$$8$$9$$x$$y$$r\cdot
NP(I)$Boundary of $\operatorname{hype}(I,r)$Minimal GeneratorsInterior
PointsExterior Points

Figure 2. Computing $\overline{(xy^{5},x^{2}y^{2},x^{4}y)^{4/3}}$ using the
Hyperrectangle algorithm (left) and Improved Hyperrectangle algorithm (right)

In general, for fixed $I$ and $r$, the two convex sets $\mathcal{C}(I,r))$ and
$\operatorname{hype}(I,r)$ where Algorithm 1 and Algorithm 2, respectively,
look for a set of generators for $\overline{I^{r}}$ are incomparable. For an
illustration consider Figure 1 in 4.2, where the set $\mathcal{C}(I,r))$ is
shaded in darker blue and Figure 2 where the set $\operatorname{hype}(I,r)$ is
the marked by the orange boundary. Note that there are no containments between
the sets $\mathcal{C}(I,r)$ and $\operatorname{hype}(I,r)$ in this example. In
general one does not expect a containment between the corresponding sets of
lattice points $\mathcal{C}(I,r))\cap{\mathbb{N}}^{d}$ and
$\operatorname{hype}(I,r)\cap{\mathbb{N}}^{d}$ either. However, the
cardinality of the former set is typically smaller than the latter. We address
this shortcoming in the next Algorithm 3.

The exponent vectors for minimal generators of $\overline{I^{r}}$ are in
$\mathcal{C}(I,r)\cap\operatorname{hype}(I,r)\cap{\mathbb{N}}^{d}$. However,
as illustrated by Figure 1 and Figure 2, the exponents for the minimal
generators of $\overline{I^{r}}$ can form a proper subset of
$\mathcal{C}(I,r))\cap\operatorname{hype}(I,r)\cap{\mathbb{N}}^{d}$.

The next variant improves on the hyperrectangle algorithm by reducing some
redundancies in the traversal of lattice points. Using the while-loop on the
final coordinate, the improved hyperrectangle algorithm stops looking for
other generators after it finds a lattice point that is inside $r\cdot NP(I)$.
Note that the improved hyperrectangle algorithm optimizes traversal of the set
$\operatorname{hype}(I,r)\cap{\mathbb{N}}^{d}$ only on the last coordinate, so
the benefits of using this algorithm over the hyperrectangle algorithm is more
apparent in low dimensional rings.

Input: the Newton polyhedron $NP(I)$ of an ideal $I$, a real number
$r\in{\mathbb{R}}_{+}$

Output: a list of monomial generators for the ideal $\overline{I^{r}}$

1 $d:=$ dimension of the polynomial ring containing $I$

2startPoints :=
$\\{\,\in\operatorname{hype}(I,r)\;|\;b_{d}=\min(\mathcal{V},d)\\}$

3Initialize generators := $\emptyset$

4for in startPoints do

5 while $\ {\bf not}$ in $r\cdot NP(I)\ {\bf and}\
b_{d}\leq\max(\mathcal{V},d)$ do

$:=(0,\dots,0,1)$ /* ‘‘move up’’ */

6

7

8 if in $r\cdot NP(I)$ then

9 generators := append(${\mathbf{x}}$, generators)

10

/* Return the possibly non minimal monomial generators */

11

Return generators.

Algorithm 3 Improved Hyperrectangle algorithm

######  Example 4.7.

Figure 2 illustrates the set of generators for the ideal
$\overline{(xy^{5},x^{2}y^{2},x^{4}y)^{4/3}}$ returned by the improved
hyperrectangle algorithm. The set of lattice points considered by this
algorithm are marked in solid yellow and purple and hollow black. The set of
generators returned by Algorithm 3 corresponds to the yellow and purple
lattice points, while the minimal generator correspond to the purple lattice
points only. Compared to Figure 2, fewer non minimal generators are returned.

###  4.3. Staircase Algorithm

The algorithms presented in the previous sections (Algorithm 1, Algorithm 2,
and Algorithm 3) have one common disadvantage in that they return possibly non
minimal sets of generators for the real powers of monomial ideals. The next
algorithm, termed the staircase algorithm, traverses lattice points near the
boundary of the Newton polyhedron. The traversal is designed so that, in the
2-dimensional case, the minimal generators are found.

A benefit of the following algorithm is to improve upon the runtime of
Algorithm 1 and Algorithm 3. Algorithm 1 is slow in practice because of
lattice points identification in step 1, while Algorithm 3 may be inefficient
because a large number of operations could be performed to check if lattice
points are in or outside $r\cdot NP(I)$. To alleviate this issue, the
staircase algorithm optimizes the traversal of lattice points on the final two
coordinates. The algorithm uses the notation in equation (4.1).

Input: the Newton polyhedron $NP(I)$ of an ideal $I$, a real number
$r\in{\mathbb{R}}_{+}$

Output: a list of monomial generators for the real power $\overline{I^{r}}$

1 Initialize generators := $\emptyset$

2$d:=$ dimension of the polynomial ring containing $I$

3if $d=1$ then

4 Return $\\{{\mathbf{x}}^{\min(\mathcal{V},1)}\\}$

5else

6 startPoints :=
$\big{\\{}\in\operatorname{hype}(I,r)\;|\;a_{d-1}=\min(\mathcal{V},d-1),\;a_{d}=\max(\mathcal{V},d)\big{\\}}$

7 for in startPoints do

8 $:=$

9 while in $\operatorname{hype}(I,r)$ do

10 if in $r\cdot NP(I)$ then

11 $:=$

$:=(0,\dots,0,1)$ /* ‘‘move down’’ */

12

13 else

14 if in $r\cdot NP(I)$ then

15 generators := append(${\mathbf{x}}$, generators)

16 $:=$

$:=(0,\dots,1,0)$ /* ``move right'' */

17

18

19

20 if in $r\cdot NP(I)$ then

21 generators := append(${\mathbf{x}}$, generators)

22

23

24

Return generators.

Algorithm 4 Staircase algorithm

######  Example 4.8.

Figure 3 shows the set of lattice points considered by the staircase algorithm
within $\operatorname{hype}(I,\frac{4}{3})$ for the ideal
$I=(xy^{5},x^{2}y^{2},x^{4}y)$. While all the lattice points along the path of
the algorithm are considered, only the minimal generators corresponding to the
purple lattice points are returned.

$0$$1$$2$$3$$4$$5$$6$$7$$8$$0$$1$$2$$3$$4$$5$$6$$7$$8$$9$startend$x$$y$$r\cdot
NP(I)$Boundary of $\operatorname{hype}(I,r)$Path of algorithmMinimal
GeneratorsInterior PointsExterior Points

Figure 3. Computing $\overline{(xy^{5},x^{2}y^{2},x^{4}y)^{4/3}}$ using the
Staircase algorithm

We are now ready to show the validity of Algorithm 4. We utilize terminology
that is consistent with the visual descriptions in Figure 3. We call the path
of the algorithm $\mathcal{P}(I,r)$ the set of values taken by the variable in
Algorithm 4 for fixed inputs $I,r$. This set is the disjoint union of two
subsets: the exterior path and the interior path defined below:

$\displaystyle\mathcal{P}_{ext}(I,r)$
$\displaystyle=\\{\in\mathcal{P}(I,r)\setminus r\cdot NP(I)\\}$
$\displaystyle\mathcal{P}_{int}(I,r)$
$\displaystyle=\\{\in\mathcal{P}(I,r)\cap r\cdot NP(I)\\}.$

######  Proposition 4.9.

If $I$ is a monomial ideal of a $d$-dimensional polynomial ring and
$d\in\\{1,2\\}$, then Algorithm 4 returns a minimal set of monomial generators
for $\overline{I^{r}}$. If $d\geq 3$ then Algorithm 4 returns a not
necessarily minimal set of monomial generators for $\overline{I^{r}}$.

###### Proof.

In the case $d=1$, every monomial ideal $J\subseteq K[x_{1}]$ is principal,
minimally generated by $x_{1}^{m}$, where $m=\min\\{a\mid x_{1}^{a}\in J\\}$.
Applying this to $J=\overline{I^{r}}$ for which case $m=\min(\mathcal{V},1)$
yields $G(\overline{I^{r}})=\\{x_{1}^{\min(\mathcal{V},1)}\\}$, i.e., the
output of Algorithm 4 in step 4.

For the case $d=2$, first notice that because of the succession of down moves
and right moves, the interior path $\mathcal{P}_{int}(I,r)$ is a disjoint
union of vertical strips of the form

$s_{a,b,c}:=\\{\gamma=(\gamma_{1},\gamma_{2})\mid\gamma_{1}=a,\,\gamma_{2}\in[b,c]\cap{\mathbb{N}}\\},$

where $b=\min\\{b^{\prime}\mid(a,b^{\prime})\in r\cdot NP(I)\\}$ by step 4 of
the algorithm; see Figure 3 for an illustration. Moreover, the interior path
contains one lattice point for each value of the $x_{2}$-coordinate in
$[\min(\mathcal{V},1),\max(\mathcal{V},1)]$ so that in the decomposition

(4.2)
$\mathcal{P}_{int}(I,r)=\bigcup_{i=\min(\mathcal{V},1)}^{e}s_{i,b_{i},c_{i}}$

we must have $c_{\min(\mathcal{V},1)}=\max(\mathcal{V},2)$ and
$b_{i}=c_{i+1}+1$ for each $i\leq e-1$, where $e$ is the maximum $x_{1}$
coordinate of any point on the interior path. In particular, if $i<j$ then the
inequality $b_{i}>c_{j}$ holds.

Let ${\mathbf{x}}^{\in}G(\overline{I^{r}})$. By 4.4 it follows that
$(a_{1},a_{2})\in\operatorname{hype}(I,r)$, so
$a_{2}\in[\min(\mathcal{V},1),\max(\mathcal{V},1)]$, and by the preceding
remarks there exists a unique point $\in\mathcal{P}_{int}(I,r)$ with
$b_{2}=a_{2}$. We claim that . If not, then $a_{1}<b_{1}$ since ${\mathbf{x}}$
is a minimal generator (i.e., lies ``left'' of ), and for this reason
$a_{2}=b_{2}\leq c_{b_{1}}<b_{a_{1}}$ (i.e lies ``below'' the strip with
$x_{1}$-coordinate $a_{1}$). Since $(a_{1},a_{2})\in r\cdot NP(I)$ and
$b_{a_{1}}=\min\\{b^{\prime}\mid(a_{1},b^{\prime})\in r\cdot NP(I)\\}$, this
yields a contradiction. We have shown that

$G(\overline{I^{r}})\subseteq\\{{\mathbf{x}}^{\mid}\in\mathcal{P}_{int}(I,r)\\}.$

In the notation of (4.2), the algorithm returns the set
$\\{x_{1}^{i}x_{2}^{b_{i}}\mid\min(\mathcal{V},1)\leq i\leq e\\}$. Each of the
monomials $x_{1}^{i}x_{2}^{j}$ with $j\in(b_{i},c_{i}]\cap{\mathbb{N}}$ are
not in $G(\overline{I^{r}})$ since they are divisible by
$x_{1}^{i}x_{2}^{b_{i}}$. Thus $G(\overline{I^{r}})$ is contained in the
returned set. Moreover, the returned set consists of minimal generators since
no two of its elements are comparable under the divisibility relation. In
fact, this proof shows that the case $d=2$ of the algorithm gives a minimal
set of generators for the ideal generated by the monomials with exponents in a
given convex set (in our application to real powers, this convex set is
$r\cdot NP(I)$). We use this to approach the higher dimensional cases.

The case $d>2$ is derived from the case $d=2$ by the following analysis. By
virtue of 4.4 we have the identity

$\displaystyle\overline{I^{r}}$ $\displaystyle=\left(\\{{\mathbf{x}}^{\mid}\
\in\operatorname{hype}(I,r)\cap r\cdot NP(I)\cap{\mathbb{N}}^{d}\right)$
$\displaystyle=\left(\sum_{\gamma\in\prod_{i=1}^{d-2}[\min(\mathcal{V},i),\,\max(\mathcal{V},i)]}x_{1}^{\gamma_{1}}\cdots
x_{d-2}^{\gamma_{d-2}}\cdot I_{\gamma,r}\right),$

where $I_{\gamma,r}:=(\\{x_{d-1}^{a}x_{d}^{b}\mid\
(\gamma_{1},\ldots,\gamma_{d-2},a,b)\in r\cdot NP(I)\\})$ is an ideal in a
2-dimensional polynomial ring. According to the case $d=2$, steps 7–19 of the
algorithm append the set $x_{1}^{\gamma_{1}}\cdots x_{d-2}^{\gamma_{d-2}}\cdot
G(I_{\gamma,r})$ to the generators list. The union of these sets generates
$\overline{I^{r}}$ by the above displayed identity. ∎

######  Example 4.10.

We give a visual illustration of using Algorithm 4 to compute the integral
closure of $I=(y^{3},y^{2}z^{5},x^{2}y^{2},x^{2}z^{3})$, that is,
$\overline{I^{1}}$ in Figure 4. In 3-dimensional space, the path of the
algorithm is a disjoint union of paths, each corresponding to an ideal in a
2-dimensional ring as shown in the proof of 4.9.

$0$$1$$2$$3$$4$$0$$1$$2$$3$$4$$0$$1$$2$$3$$4$$5$start pointsend
points$x$$y$Boundary of $r\cdot NP(I)$Interior PointsMinimal
GeneratorsExterior Points

Figure 4. Computing generators for
$\overline{(y^{3},y^{2}z^{5},x^{2}y^{2},x^{2}z^{3})}$ using the Staircase
algorithm

##  5\. Continuity and jumping numbers for exponentiation

In this section we analyze how the real powers of monomial ideals vary with
the exponent. To be precise, for a fixed monomial ideal $I$ we consider
continuity properties for the exponentiation function of base $I$

$\exp:{\mathbb{R}}_{+}\to\mathcal{T},\quad\exp(r)=\overline{I^{r}}$

whose domain is ${\mathbb{R}}_{+}$ with its Euclidean topology and whose
codomain is the set $\mathcal{T}=\\{\overline{I^{r}}\mid
r\in{\mathbb{R}}_{+}\\}$ endowed with the discrete topology.

We start with two elementary properties enjoyed by the family of real powers
of the fixed ideal.

######  Lemma 5.1.

If $I$ is a monomial ideal and $r,s\in{\mathbb{R}}_{+}$ then

  1. (1)

if $s\geq r\geq 0$, then the containment
$\overline{I^{s}}\subseteq\overline{I^{r}}$ holds,

  2. (2)

$\overline{I^{s}}\cdot\overline{I^{r}}\subseteq\overline{I^{s+r}}$.

###### Proof.

Assertion (1) is clear from 3.1. To clarify assertion (2), note that monomials
in $\overline{I^{s}}\cdot\overline{I^{r}}$ correspond to lattice points in the
Minkowski sum

$s\cdot NP(I)+r\cdot NP(I)=(s+r)\cdot NP(I).$

∎

Part (2) of 5.1 shows that the real powers of a fixed monomial ideal form a
graded family, although this terminology is more commonly used for families
indexed by a discrete set. Property (1) of 5.1 allows to define for each
$r\in{\mathbb{R}}$ the monomial ideal

$\overline{I^{>r}}=\bigcup_{s>r}\overline{I^{s}}.$

We show that this ideal can be understood as a limit in $\mathcal{T}$, meaning
that a sequence of real powers of $I$ where the exponents approach a real
number $r$ from the right must stabilize to $\overline{I^{>r}}$.

######  Proposition 5.2.

Let $I$ be a monomial ideal and let $\\{t_{n}\\}_{n\in{\mathbb{N}}}$ be a non-
increasing sequence of non-negative real numbers with
$r=\lim_{n\to\infty}t_{n}$. Then $\overline{I^{t_{n}}}=\overline{I^{>r}}$ for
$n$ sufficiently large.

###### Proof.

A non-increasing sequence of non-negative numbers
$\\{t_{n}\\}_{n\in{\mathbb{N}}}$ gives an ascending chain of ideals
$\overline{I^{t_{0}}}\subseteq\overline{I^{t_{1}}}\subseteq\cdots\subseteq\overline{I^{r}}$
by 5.1 (1). Since the polynomial ring is Noetherian, any such chain must in
fact stabilize, i.e. there exists $N\gg 0$ such that
$\overline{I^{t_{n}}}=\overline{I^{t_{m}}}$ for $m,n\geq N$. We show that the
stable value of this chain is $\overline{I^{>r}}$. Indeed, from the definition
of $\overline{I^{>r}}$ one deduces the containment

$\overline{I^{t_{N}}}=\bigcup_{n=0}^{\infty}\overline{I^{t_{n}}}\subseteq\bigcup_{s>r}\overline{I^{s}}=\overline{I^{>r}}.$

Conversely, for each $s>r$, there exists $n\geq N$ such that $s>t_{n}$, hence
one has the containments
$\overline{I^{s}}\subseteq\overline{I^{t_{N}}}=\overline{I^{t_{n}}}$ for all
$s>r$ and consequently $\overline{I^{t_{N}}}\supseteq\overline{I^{>r}}$. ∎

To distinguish those real numbers $r$ for which the function
$\exp:{\mathbb{R}}_{+}\to\mathcal{T},\exp(r)=\overline{I^{r}}$ is right
discontinuous, we term them jumping numbers.

######  Definition 5.3.

A jumping number for $I$ is a real number $r\in{\mathbb{R}}_{+}$ for which the
real powers of $I$ are not equal to $\overline{I^{r}}$ when we approach $r$
from the right, i.e.

$\overline{I^{r}}\neq\overline{I^{>r}}.$

######  Example 5.4.

0 is a jumping number for any monomial ideal since $\overline{I^{0}}=R$ but
$\overline{I^{r}}$ is a proper ideal for any $r>0$.

######  Example 5.5.

For $I=(x^{4},x^{2}y,xy^{3})$ we have that $\frac{1}{3}$ is not a jumping
number while $\frac{1}{2}$ is a jumping number. This is because for small
values of $\varepsilon>0$ there is an equality

$\frac{1}{3}\cdot
NP(I)\cap{\mathbb{N}}^{2}=\left(\frac{1}{3}+\varepsilon\right)\cdot
NP(I)\cap{\mathbb{N}}^{2},$

while

$\frac{1}{2}\cdot
NP(I)\cap{\mathbb{N}}^{2}\neq\left(\frac{1}{2}+\varepsilon\right)\cdot
NP(I)\cap{\mathbb{N}}^{2}$

because the point $(2,0)$ belongs to the leftmost set but not the rightmost.
In fact, for the ideal $I$ in this example, we have
$(x^{2},xy)=\overline{I^{1/3}}=\overline{I^{>1/3}}=\overline{I^{1/2}}\neq\overline{I^{>1/2}}=(x^{3},xy)$.

$0$$1$$2$$3$$0$$1$$2$$3$$x$$y$$0$$1$$2$$3$$0$$1$$2$$3$$x$$y$

Figure 5. Comparing $\frac{1}{3}\cdot NP(I)$ and $\frac{1}{2}\cdot NP(I)$

To verify that right continuity is a special characteristic to study, we show
that the exponentiation function is a left continuous function.

Towards this end recall that any polyhedron admits a description as a finite
intersection of half spaces. We term the linear inequalities describing a
polyhedron as an intersection of half spaces its bounding inequalities. In
particular, if $I$ is a monomial ideal in a polynomial ring of dimension $d$
then $NP(I)$ is a lattice polyhedron, hence there exist a $d\times s$ matrix
$A$ with entries in ${\mathbb{N}}$ and a vector
${\mathbf{c}}\in{\mathbb{N}}^{d}$ such that

(5.1) $NP(I)=\\{{\mathbf{x}}\in{\mathbb{R}}_{+}^{d}\mid
A{\mathbf{x}}\geq{{\mathbf{c}}}\\}.$

In (5.1), if $A=[a_{ij}]$, we will further assume that we have
$\gcd(a_{i1},\ldots,a_{is},c_{i})=1$ for each $1\leq i\leq d$. Moreover,
scaling the Newton polyhedron amounts to scaling the constant term of the
bounding inequalities, that is,

$r\cdot NP(I)=\\{{\mathbf{x}}\in{\mathbb{R}}_{+}^{d}\mid A{\mathbf{x}}\geq
r\cdot{{\mathbf{c}}}\\}.$

######  Proposition 5.6.

The function $\exp:{\mathbb{R}}_{+}\to\mathcal{T},\exp(r)=\overline{I^{r}}$ is
left continuous.

###### Proof.

Fix $r\in{\mathbb{R}}_{+}$ and consider the set
$A_{r}={\mathbb{N}}^{d}\setminus r\cdot NP(I)$. Observe that each point
${\mathbf{p}}\in A_{r}$ lies at a positive Euclidean distance from any point
in $r\cdot NP(I)$. Indeed in the notation of (5.1), writing i for the $i$-th
row of $A$ we have ${}_{i}\cdot{\mathbf{p}}<rc_{i}$ for at least one $1\leq
i\leq d$ and thus the distance from ${\mathbf{p}}$ to the hyperplane of
equation ${}_{i}\cdot{\mathbf{x}}=rc_{i}$ is
$d_{i}=(rc_{i}-_{i}\cdot{\mathbf{p}})/\sqrt{{}_{i}\cdot_{i}}>0$. In
particular, since ${}_{i}\cdot{\mathbf{p}}\in{\mathbb{N}}$, it follows that
$d_{i}\geq\delta_{i}:=(rc_{i}-{\rm prec}(rc_{i}))/\sqrt{{}_{i}\cdot_{i}}>0$,
where ${\rm prec}(u)$ is the largest integer strictly smaller than $u$. Taking
$\Delta=\min_{1\leq i\leq d}\delta_{i}$ we conclude that any ${\mathbf{p}}\in
A_{r}$ lies at distance at least $\Delta>0$ from any point in $NP(I)$.

Since each $\delta_{i}$ is a left continuous function of $r$, it follows that
there exists $\varepsilon_{0}>0$ such that for any
$0<\varepsilon<\varepsilon_{0}$ each point ${\mathbf{p}}\in A_{r}$ lies at a
positive Euclidean distance from any point in $(r-\varepsilon)\cdot NP(I)$ as
well. Equivalently we have $A_{r}\cap(r-\varepsilon)\cdot NP(I)=\emptyset$
which yields $A_{r}=A_{r-\varepsilon}$ and thus $r\cdot
NP(I)\cap{\mathbb{N}}^{d}=(r-\varepsilon)\cdot NP(I)\cap{\mathbb{N}}^{d}$ and
$\overline{I^{r}}=\overline{I^{r-\varepsilon}}$ for
$0<\varepsilon<\varepsilon_{0}$. ∎

We now show that the real exponentiation function of a monomial ideal is a
step function.

######  Corollary 5.7.

Let $j<j^{\prime}$ be two consecutive jumping numbers for $I$. Then the
function $\exp:{\mathbb{R}}_{+}\to\mathcal{T},\exp(r)=\overline{I^{r}}$ is
constant on $(j,j^{\prime}]$ and
$\overline{I^{j}}\neq\overline{I^{j^{\prime}}}$.

###### Proof.

Since $j<j^{\prime}$ are consecutive jumping numbers, meaning there is no
jumping number in $(j,j^{\prime})$, the exponentiation function is continuous
on $(j,j^{\prime}]$ by a combination of 5.2 and 5.6, and left continuity at
$j^{\prime}$. Since $\mathcal{T}$ carries the discrete topology, this
continuity is equivalent to the function being constant on $(j,j^{\prime}]$.
However, the exponentiation function is right discontinuous at $j$ by the
definition of jumping number, thus $\overline{I^{j}}$ is distinct from the
common value of the exponentiation function on $(j,j^{\prime}]$, that is,
$\overline{I^{j}}\neq\overline{I^{j^{\prime}}}$. ∎

Our next aim is to show that the jumping numbers for monomial ideals are
rational. Utilizing the notation in (5.1) and setting i to be the $i$-th row
of the matrix $A$ therein, the facets of the Newton polyhedron are supported
on hyperplanes $H_{i}$ with equation ${}_{i}{\mathbf{x}}=c_{i}$. Each facet
$F_{i}$ of $NP(I)$ is thus cut out by a system formed by one equation and
several inequalities of the form

(5.2)
$F_{i}=\left\\{{\mathbf{x}}\mid_{i}\cdot{\mathbf{x}}=c_{i},_{j}\cdot{\mathbf{x}}\geq
c_{j}\text{ for }1\leq j\leq d,j\neq i\right\\}.$

######  Proposition 5.8.

Given a monomial ideal $I$ with facets $F_{i},1\leq i\leq s$ for $NP(I)$
described as in (5.2) above, the following are equivalent:

  1. (1)

$r\in{\mathbb{R}}_{+}$ is a jumping number for $I$;

  2. (2)

for some $1\leq i\leq s$ such that $c_{i}\neq 0$ there exists a lattice point
${\mathbf{p}}\in r\cdot F_{i}\cap{\mathbb{N}}^{d}$;

  3. (3)

for some $1\leq i\leq s$ such that $c_{i}\neq 0$ there exists an integer
solution to the system of equations and inequalities that describes $r\cdot
F_{i}$, namely

(5.3) $\begin{cases}{}_{i}\cdot{\mathbf{x}}=rc_{i}\\\
{}_{j}\cdot{\mathbf{x}}\geq rc_{j}\text{ for }1\leq j\leq d,j\neq
i.\end{cases}$

###### Proof.

$(2)\Leftrightarrow(3)$ is clear.

$(1)\Rightarrow(2)$ We show the contrapositive. Assume that
$r\in{\mathbb{R}}_{+}$ is such that the union of the facets of $r\cdot F_{i}$
of $r\cdot NP(I)$ corresponding to $c_{i}\neq 0$ contains no lattice point.
Note that

$r\cdot NP(I)\setminus(r+\varepsilon)\cdot
NP(I)\subseteq\\{{\mathbf{x}}\mid_{i}\cdot{\mathbf{x}}\in[rc_{i},(r+\varepsilon)c_{i})\text{
for some }i\text{ with }c_{i}\neq 0\\}.$

Moreover there is at least one $1\leq i\leq d$ so that $c_{i}\neq 0$ since
$I\neq R$. Taking $\varepsilon<\varepsilon_{0}:=\min_{c_{i}\neq
0}\\{(\rm{next}(rc_{i})-rc_{i})/c_{i}\\}$, where $\rm{next}(rc_{i})$ is the
smallest integer strictly larger than $rc_{i}$, one can ensure that
$[rc_{i},(r+\varepsilon)c_{i})\cap{\mathbb{N}}\subseteq\\{rc_{i}\\}$ whenever
$c_{i}\neq 0$. This means that any possible lattice point ${\mathbf{t}}$ in
$r\cdot NP(I)\setminus(r+\varepsilon)\cdot NP(I)$ satisfies
${}_{i}\cdot{\mathbf{t}}=rc_{i}$ for some $c_{i}\neq 0$. Thus we see that
${\mathbf{t}}$ lies on a facet $F_{i}$ which contains no lattice points by
assumption. Thus there are no lattice points in $r\cdot
NP(I)\setminus(r+\varepsilon)\cdot NP(I)$.

It follows that $\overline{I^{r}}=\overline{I^{r+\varepsilon}}$ for
$0<\varepsilon<\varepsilon_{0}$ and thus $r$ is not a jumping number for $I$.

$(3)\Rightarrow(1)$ Let $\mathbf{p}\in{\mathbb{N}}^{d}$ be an integer solution
to (5.3). Since this implies ${\mathbf{p}}\in r\cdot F_{i}\subseteq r\cdot
NP(I)$, we see that ${\mathbf{x}}^{\mathbf{p}}\in\overline{I^{r}}$. Since
${\mathbf{p}}$ attains equality in the first equation of (5.3) it follows that
${\mathbf{p}}$ satisfies ${}_{i}\cdot{\mathbf{p}}<(r+\varepsilon)c_{i}$ for
any $\varepsilon>0$. (This uses $c_{i}\neq 0$.) Thus we conclude
${\mathbf{p}}\not\in(r+\varepsilon)\cdot NP(I)$ and
${\mathbf{x}}^{\mathbf{p}}\notin\overline{I^{r+\epsilon}}$ for all
$\epsilon>0$ and therefore ${\mathbf{x}}^{\mathbf{p}}\notin\overline{I^{>r}}$.
Consequently $r$ is a jumping number. ∎

From the above characterization we obtain that jumping numbers control the
behavior of all real powers of a given monomial ideal and are all rational
numbers.

######  Theorem 5.9.

Let $I$ be a monomial ideal.

  1. (1)

All jumping numbers for $I$ are rational.

  2. (2)

All distinct real powers of $I$ are given by rational exponents, i.e., for
each $r\in{\mathbb{R}}_{+}$ there exists $r^{\prime}\in{\mathbb{Q}}$ so that
$\overline{I^{r}}=\overline{I^{r^{\prime}}}$. Moreover $r^{\prime}$ can be
taken to be a jumping number for $I$.

  3. (3)

If $r$ is a jumping number of $I$ then $nr$ is a jumping number for all
$n\in{\mathbb{N}}$.

  4. (4)

If $\mathbf{v}$ is a vertex of $NP(I)$, then for all $n\in{\mathbb{N}}$ the
number $r_{n}=\frac{n}{\gcd(v_{1},\cdots,v_{d})}$ is a jumping number of $I$ .

  5. (5)

The set of jumping numbers can be written as a finite union of scaled monoids
$\mathcal{J}=\bigcup_{c_{i}\neq 0}\frac{1}{c_{i}}S_{i}.$ Here each $S_{i}$ is
a submonoid of the numerical semigroup generated by the entries of the $i$-th
row of the matrix $A$ in (5.1) and $c_{i}$ are the components of the vector
${\mathbf{c}}$ in (5.1).

###### Proof.

(1) follows since 5.8 (3) yields that there is an integer solution
${\mathbf{p}}$ to an equation of the form ${}_{i}\cdot{\mathbf{p}}=rc_{i}$
where i is a row of the matrix $A$ in (5.1) and $c_{i}\neq 0$. Since the
entries of ${}_{i},{\mathbf{p}}$, and $c_{i}$ are natural numbers, this gives
$r\in{\mathbb{Q}}$.

(2) If $r\in{\mathbb{Q}}_{+}$ is a jumping number, set $r^{\prime}=r$. If $r$
is not a jumping number, let

$r^{\prime}=\inf\\{u\mid u>r\text{ and }u\text{ is a jumping number for
}I\\}.$

Notice first that $r^{\prime}$ is in fact the minimum of the set above,
equivalently $r^{\prime}\in{\mathbb{Q}}$ is a jumping number for $I$. Indeed,
if this is not the case, then there is a sequence of pairwise distinct jumping
numbers $\\{u_{n}\\}_{n\in{\mathbb{N}}}$ converging to $r^{\prime}$ from the
right. Since we have assumed $r^{\prime}$ is not a jumping number, the
exponential function with base $I$ is right continuous at $r^{\prime}$, thus
it must be the case that $\overline{I^{u_{n}}}=\overline{I^{r^{\prime}}}$ for
$n\gg 0$. This contradicts that the numbers $u_{n}$ are distinct jumping
numbers, since distinct jumping numbers yield distinct real powers by 5.7.
Another application of 5.7 together with the observation that $r$ is not a
jumping number yields that the exponentiation function is constant on
$[r,r^{\prime}]$, thus we conclude there is an equality
$\overline{I^{r}}=\overline{I^{r^{\prime}}}$.

(3) follows since the condition on integer solutions to the system (5.3) in
5.8 is preserved upon scaling the system by any natural number.

(4) Each vertex ${\mathbf{v}}$ of $NP(I)$ furnishes an integer solution to the
system of (in)equalities (5.3) corresponding to each facet $F_{i}$ such that
${\mathbf{v}}\in F_{i}$. Scaling by $r_{n}$ we see that
$r_{n}\cdot{\mathbf{v}}\in{\mathbb{N}}^{d}$ is an integer solution to the
analogous system

$\begin{cases}{}_{i}\cdot{\mathbf{x}}=r_{n}c_{i},\\\
{}_{j}\cdot{\mathbf{x}}\geq r_{n}c_{j}\text{ for }1\leq j\leq d,i\neq
j.\end{cases}$

5.8 yields that $r_{n}$ is a jumping number for $I$.

For (5), for each $1\leq i\leq d$, let $a_{ij}\in{\mathbb{N}}$ be the entries
in the $i$-th row of the matrix $A$ in (5.1) and $c_{i}$ the entries of
${\mathbf{c}}$. For each $i$ with $c_{i}\neq 0$ set

$S_{i}=\left\\{rc_{i}\mid\exists
x_{1},\ldots,x_{d}\in{\mathbb{N}}\cup\\{0\\}\text{
s.t.}\sum_{j=1}^{d}a_{ij}x_{j}=rc_{i},\sum_{j=1}^{d}a_{lj}x_{j}\geq
rc_{l}\text{ for }l\neq i\right\\}.$

It is clear that $S_{i}\subset{\mathbb{N}}\cup\\{0\\}$. Moreover $S_{i}$ is a
monoid as $0\in S_{i}$ and $rc_{i},r^{\prime}c_{i}\in S_{i}$ imply
$(r+r^{\prime})c_{i}\in S_{i}$ by summing the respective (in)equalities. The
existence of a non-negative integer solution to the equation
$\sum_{j=1}^{d}a_{ij}x_{j}=rc_{i}$ implies that $rc_{i}$ belongs to the
numerical semigroup $M_{i}$ generated by the integers $a_{ij}$ for $1\leq
j\leq d$, thus $S_{i}\subseteq M_{i}$. With this notation, 5.8 can be
rephrased to say that the set of jumping numbers for $I$ is

$\mathcal{J}=\bigcup_{c_{i}\neq 0}\frac{1}{c_{i}}S_{i}.$

∎

In regards to item (1) of 5.9 we observe that every non-negative rational
number is a jumping number for some monomial ideal. Indeed if $r=\frac{p}{q}$
with $p,q\in{\mathbb{N}},q\neq 0$ then $r$ is a jumping number of
$I=(x_{1}^{q})$.

Item (2) of 5.9 yields a new description for the image of the exponentiation
function with base $I$

$\mathcal{T}=\\{\overline{I^{r}}\mid r\in{\mathbb{Q}}\text{ is a jumping
number for }I\\}.$

Moreover, the elements of the set $\mathcal{T}$ listed above are pairwise
distinct by 5.7.

We end with a worked out example which illustrates the jumping numbers and
real powers of a particular monomial ideal using the criterion in 5.8.

######  Example 5.10.

The monomial ideal $I=(x^{9},x^{4}y^{3},x^{2}y^{5},y^{8})$ has Newton
polyhedron depicted in Figure 6 with vertices at $(9,0),(4,3),(2,5),(0,8)$.

$0$$1$$2$$3$$4$$5$$6$$7$$8$$9$$10$$0$$1$$2$$3$$4$$5$$6$$7$$8$$9$$x$$y$$NP(I)$$p\geq
0$$3p+2q\geq 16$$p+q\geq 7$$3p+5q\geq 27$$q\geq 0$ Figure 6. The Newton
polyhedron of $(x^{9},x^{4}y^{3},x^{2}y^{5},y^{8})$

We show that the jumping numbers of $I$ are the elements of the following set

(5.4) $\mathcal{J}=\left\\{0,\frac{i}{7},\frac{j}{16},\frac{k}{27}\mid i\geq
2,j\in\\{2,4,6\\}\text{ or }j\geq 8,k\in\\{3,6,9,11,12,14,15\\}\text{ or
}k\geq 17\right\\}.$

The faces of the Newton polyhedron
${\color[rgb]{0,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,1}\pgfsys@color@cmyk@stroke{1}{0}{0}{0}\pgfsys@color@cmyk@fill{1}{0}{0}{0}F_{1}},{\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}F_{2}},{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}F_{3}},{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}F_{4}},{\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}F_{5}}$
are shown in Figure 6 together with the corresponding bounding inequalities
for $NP(I)$. Putting these inequalities in the form of (5.1) yields

$\begin{bmatrix}1&0\\\ 3&2\\\ 1&1\\\ 3&5\\\
0&1\end{bmatrix}\cdot\begin{bmatrix}p\\\ q\end{bmatrix}\geq\begin{bmatrix}0\\\
16\\\ 7\\\ 27\\\ 0\end{bmatrix}.$

5.9 (5) says that the jumping numbers depend on the following three monoids:

$\displaystyle S_{2}$ $\displaystyle=$ $\displaystyle\\{16r\mid\exists
p,q\in{\mathbb{N}}\cup\\{0\\}\text{ s.t. }p\geq 0,3p+2q=16r,p+q\geq
7r,3p+5q\geq 27r,q\geq 0\\},$ $\displaystyle S_{3}$ $\displaystyle=$
$\displaystyle\\{7r\mid\exists p,q\in{\mathbb{N}}\cup\\{0\\}\text{ s.t. }p\geq
0,3p+2q\geq 16r,p+q=7r,3p+5q\geq 27r,q\geq 0\\},$ $\displaystyle S_{4}$
$\displaystyle=$ $\displaystyle\\{27r\mid\exists
p,q\in{\mathbb{N}}\cup\\{0\\}\text{ s.t. }p\geq 0,3p+2q\geq 16r,p+q\geq
7r,3p+5q=27r,q\geq 0\\}.$

Denote ${\mathbb{N}}_{0}={\mathbb{N}}\cup\\{0\\}$. It turns out that
$S_{2}=2{\mathbb{N}}_{0}+9{\mathbb{N}}_{0}$,
$S_{3}=2{\mathbb{N}}_{0}+3{\mathbb{N}}_{0}$, and
$S_{4}=3{\mathbb{N}}_{0}+11{\mathbb{N}}_{0}+19{\mathbb{N}}_{0}$. The set of
jumping numbers is

$\mathcal{J}=\frac{1}{16}S_{2}\cup\frac{1}{7}S_{3}\cup\frac{1}{27}S_{4},$

Writing the the elements of each semigroup $S_{1},S_{2},S_{3}$ explicitly
yields the set displayed in equation (5.4) above.

Below we list the rational powers of $I$ for exponents $r\in(0,1]$. The
generators have been color coded based on the bounded edge of the Newton
polyhedron that is giving rise to change in generator(s) cf. 5.8 (2). Refer to
the legend in Figure 6 for the color corresponding to each edge.

$\overline{I^{r}}=\begin{cases}(y,x)&r\in(0,\frac{1}{9}]\\\
(y,{\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}x^{2}})&r\in(\frac{1}{9},\frac{1}{8}]\\\
({\color[rgb]{0,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,1}\pgfsys@color@cmyk@stroke{1}{0}{0}{0}\pgfsys@color@cmyk@fill{1}{0}{0}{0}y^{2}},{\color[rgb]{0,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,1}\pgfsys@color@cmyk@stroke{1}{0}{0}{0}\pgfsys@color@cmyk@fill{1}{0}{0}{0}xy},x^{2})&r\in(\frac{1}{8},\frac{2}{9}]\\\
(y^{2},xy,{\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}x^{3}})&r\in(\frac{2}{9},\frac{2}{8}]\\\
({\color[rgb]{0,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,1}\pgfsys@color@cmyk@stroke{1}{0}{0}{0}\pgfsys@color@cmyk@fill{1}{0}{0}{0}y^{3}},xy,x^{3})&r\in(\frac{2}{8},\frac{2}{7}]\\\
(y^{3},{\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}xy^{2}},{\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}x^{2}y},x^{3})&r\in(\frac{2}{7},\frac{3}{9}]\\\
(y^{3},xy^{2},x^{2}y,{\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}x^{4}})&r\in(\frac{3}{9},\frac{3}{8}]\\\
({\color[rgb]{0,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,1}\pgfsys@color@cmyk@stroke{1}{0}{0}{0}\pgfsys@color@cmyk@fill{1}{0}{0}{0}y^{4}},xy^{2},x^{2}y,x^{4})&r\in(\frac{3}{8},\frac{11}{27}]\\\
(y^{4},xy^{2},{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}x^{3}y},x^{4})&r\in(\frac{11}{27},\frac{3}{7}]\\\
(y^{4},{\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}xy^{3}},{\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}x^{2}y^{2}},x^{3}y,x^{4})&r\in(\frac{3}{7},\frac{4}{9}]\\\
(y^{4},xy^{3},x^{2}y^{2},x^{3}y,{\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}x^{5}})&r\in(\frac{4}{9},\frac{4}{8}]\\\
({\color[rgb]{0,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,1}\pgfsys@color@cmyk@stroke{1}{0}{0}{0}\pgfsys@color@cmyk@fill{1}{0}{0}{0}y^{5}},xy^{3},x^{2}y^{2},x^{3}y,x^{5})&r\in(\frac{4}{8},\frac{14}{27}]\\\
(y^{5},xy^{3},x^{2}y^{2},{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}x^{4}y},x^{5})&r\in(\frac{14}{27},\frac{5}{9}]\\\
(y^{5},xy^{3},x^{2}y^{2},x^{4}y,{\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}x^{6}})&r\in(\frac{5}{9},\frac{9}{16}]\\\
(y^{5},{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}xy^{4}},x^{2}y^{2},x^{4}y,x^{6})&r\in(\frac{9}{16},\frac{4}{7}]\\\
(y^{5},xy^{4},{\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}x^{2}y^{3}},{\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}x^{3}y^{2}},x^{4}y,x^{6})&r\in(\frac{4}{7},\frac{5}{8}]\\\
({\color[rgb]{0,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,1}\pgfsys@color@cmyk@stroke{1}{0}{0}{0}\pgfsys@color@cmyk@fill{1}{0}{0}{0}y^{6}},xy^{4},x^{2}y^{3},x^{3}y^{2},x^{4}y,x^{6})&r\in(\frac{5}{8},\frac{17}{27}]\\\
(y^{6},xy^{4},x^{2}y^{3},x^{3}y^{2},{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}x^{5}y},x^{6})&r\in(\frac{17}{27},\frac{6}{9}]\\\
(y^{6},xy^{4},x^{2}y^{3},x^{3}y^{2},x^{5}y,{\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}x^{7}})&r\in(\frac{6}{9},\frac{11}{16}]\\\
(y^{6},{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}xy^{5}},x^{2}y^{3},x^{3}y^{2},x^{5}y,x^{7})&r\in(\frac{11}{16},\frac{19}{27}]\\\
(y^{6},xy^{5},x^{2}y^{3},{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}x^{4}y^{2}},x^{5}y,x^{7})&r\in(\frac{19}{27},\frac{5}{7}]\\\
(y^{6},xy^{5},{\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}x^{2}y^{4}},{\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}x^{3}y^{3}},x^{4}y^{2},x^{5}y,x^{7})&r\in(\frac{5}{7},\frac{20}{27}]\\\
(y^{6},xy^{5},x^{2}y^{4},x^{3}y^{3},x^{4}y^{2},{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}x^{6}y},x^{7})&r\in(\frac{20}{27},\frac{3}{4}]\\\
({\color[rgb]{0,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,1}\pgfsys@color@cmyk@stroke{1}{0}{0}{0}\pgfsys@color@cmyk@fill{1}{0}{0}{0}y^{7}},xy^{5},x^{2}y^{4},x^{3}y^{3},x^{4}y^{2},x^{6}y,x^{7})&r\in(\frac{3}{4},\frac{7}{9}]\\\
(y^{7},xy^{5},x^{2}y^{4},x^{3}y^{3},x^{4}y^{2},x^{6}y,{\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}x^{8}})&r\in(\frac{7}{9},\frac{13}{16}]\\\
(y^{7},{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}xy^{6}},x^{2}y^{4},x^{3}y^{3},x^{4}y^{2},x^{6}y,x^{8})&r\in(\frac{13}{16},\frac{22}{27}]\\\
(y^{7},xy^{6},x^{2}y^{4},x^{3}y^{3},{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}x^{5}y^{2}},x^{6}y,x^{8})&r\in(\frac{22}{27},\frac{23}{27}]\\\
(y^{7},xy^{6},x^{2}y^{4},x^{3}y^{3},x^{5}y^{2},{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}x^{7}y},x^{8})&r\in(\frac{23}{27},\frac{6}{7}]\\\
(y^{7},xy^{6},{\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}x^{2}y^{5}},{\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}x^{3}y^{4}},{\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}x^{4}y^{3}},x^{5}y^{2},x^{7}y,x^{8})&r\in(\frac{6}{7},\frac{7}{8}]\\\
({\color[rgb]{0,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,1}\pgfsys@color@cmyk@stroke{1}{0}{0}{0}\pgfsys@color@cmyk@fill{1}{0}{0}{0}y^{8}},xy^{6},x^{2}y^{5},x^{3}y^{4},x^{4}y^{3},x^{5}y^{2},x^{7}y,x^{8})&r\in(\frac{7}{8},\frac{8}{9}]\\\
(y^{8},xy^{6},x^{2}y^{5},x^{3}y^{4},x^{4}y^{3},x^{5}y^{2},x^{7}y,{\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}x^{9}})&r\in(\frac{8}{9},\frac{25}{27}]\\\
(y^{8},xy^{6},x^{2}y^{5},x^{3}y^{4},x^{4}y^{3},{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}x^{6}y^{2}},x^{7}y,x^{9})&r\in(\frac{25}{27},\frac{15}{16}]\\\
(y^{8},{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}xy^{7}},x^{2}y^{5},x^{3}y^{4},x^{4}y^{3},x^{6}y^{2},x^{7}y,x^{9})&r\in(\frac{15}{16},\frac{26}{27}]\\\
(y^{8},xy^{7},x^{2}y^{5},x^{3}y^{4},x^{4}y^{3},x^{6}y^{2},{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}x^{8}y},x^{9})&r\in(\frac{26}{27},1]\\\
\end{cases}$

Acknowledgement. We warmly thank the anonymous referees for providing careful
and detailed comments which greatly improved the paper and in particular for
their contributions to 2.5 and 5.8.

This work was completed in the framework of the 2020 Polymath Jr. program
https://geometrynyc.wixsite.com/polymathreu. The second author was supported
by the NSF RTG grant in algebra and combinatorics at the University of
Minnesota DMS–1745638. The fourth author was supported by NSF DMS–2101225.

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