Syzygies over the Polytope Semiring
Abstract: Tropical geometry and its applications indicate a “theory of syzygies” over polytope semirings. Taking cue from this indication, we study a notion of syzygies over the polytope semiring. We begin our exploration with the concept of Newton basis, an analogue of Gröbner basis that captures the image of an ideal under the Newton polytope map. The image of a graded ideal under the Newton polytope map is a graded sub-semimodule of the polytope semiring. Analogous to the Hilbert series, we define the notion of Newton-Hilbert series that encodes the rank of each graded piece of . We prove the rationality of the Newton-Hilbert series for sub-semimodules that satisfy a property analogous to Cohen-Macaulayness. We define the notions of regular sequence of polytopes and syzygies of polytopes. We show an analogue of the Koszul property characterizing the syzygies of a regular sequence of polytopes.
The concept of Newton polytope [6, Chapter 4] of a Laurent polynomial is a widely studied and useful concept that in many situations, captures important properties of the hypersurface defined by . For an arbitrary subvariety of projective space, this construction is usually generalized to the Chow polytope [6, Chapter 4] associated to this subvariety. We undertake a generalization of the Newton polytope in a different direction: given an ideal of the polynomial ring where is a field of infinite cardinality, we associate a space of polytopes to it. This space is the sub-semimodule of the polytope semiring generated by the Newton polytope of every element in .
Newton Basis and Newton-Hilbert Series: Let be a field of infinite cardinality. Let be the polytope semiring whose elements are lattice polytopes with vertices in , addition in this semiring is given by convex hull and multiplication is given by Minkowski sum along with the element that is the additive identity and satisfies for all polytopes in . The only vertex of the “polytope” is itself. Depending on the context, we sometimes denote the polytope semiring simply by .
Let be the map taking a non-zero polynomial to its Newton polytope and zero to the element . We refer to this map as the Newton polytope map. Given an ideal of , the image is a sub-semimodule of (see Lemma 2.1) that we call the Newton semimodule of . A Newton basis of is a subset of such that the set generates .
Consider the irrelevant ideal of . The Newton semimodule of consists of all polytopes in except the lattice polytope defined by the origin. The set is a Newton basis for . More generally, for a monomial ideal the monomial minimal generating set forms a Newton basis for . We shall see in Section 6, Example 6.4 that an ideal need not have a finite Newton basis. ∎
The polytope semiring carries a natural -grading where the -th graded piece consists of all polytopes in that are contained in the hyperplane (see Definition 2.2 for a precise definition of grading on ). Suppose that is a -graded ideal then is naturally a -graded sub-semimodule of and each graded piece is a semigroup under the operation . This semigroup has a unique minimal generating set and this set has finite cardinality.
Let be a graded sub-semimodule of . The -th Newton-Hilbert coefficient of is defined as follows.
where is the number of minimal generators of , the -th graded piece, as a semigroup (under the operation ). This notion is in the same spirit as the Barvinok rank of a matrix .
The Newton-Hilbert Series of is the following formal power series.
If , we denote simply as . In this context, a natural question is whether the Newton-Hilbert series and the Hilbert series are equal. For a monomial ideal , we have . But in general we only have “upper-semicontinuity” i.e.,
for all (Corollary 2.6).
where and are the -th Newton-Hilbert and Hilbert coefficient of respectively.
Let be the toric ideal associated to the root lattice of type . The Newton semimodule is a graded sub-semimodule and is minimally generated as a semigroup by the polytopes where is the convex hull of and and is the -th standard basis vector of . In particular, the first Newton-Hilbert coefficient of is . Hence, unlike the case of commutative algebras, the -th Newton-Hilbert coefficient of a sub-semimodule is not necessarily upper bounded by the corresponding Newton-Hilbert coefficient of the polytope semiring. ∎
A general problem on Newton-Hilbert series is the following:
Classify power series that can occur as the Newton-Hilbert series of sub-semimodules of the polytope semiring.
We show the rationality of Newton-Hilbert series for sub-semimodules satisfying a property analogous to Cohen-Macaulayness (see Definition 3.6).
(Rationality of Newton-Hilbert Series) Let be a Cohen-Macaulay graded sub-semimodule of , then the Newton-Hilbert series of is a rational function.
The Giansiracusa brothers  study the notion of Hilbert polynomial of a tropical variety. For the toric ideal associated to the root lattice , their Hilbert polynomial is different from the polynomial underlying the Newton-Hilbert series. In fact, the tropical Hilbert polynomial defined in  coincides with the Hilbert polynomial of the underlying ideal.
Regular Sequences, Syzygies and Koszul Property for Polytopes: We formulate homological notions such as regular sequences and syzygies over polytopes. The main challenge in formulating these notions over semirings is the lack of additive inverse. For instance, the concept of “kernel of a map” is not well-defined.
For polytopes , let denote the sub-semimodule in the polytope semiring generated by .
Regular Sequences: A sequence of polytopes is called regular if for every from two to , we have the following property:
Our definition of regular sequences is motivated by the following definition of a regular sequence of elements in an integral domain: elements in an integral domain form a regular sequence if for every from two to , the element is a non-zero divisor of i.e., for every .
We construct examples of regular and non-regular sequences in Section 4. In particular, the sequence of coordinate points is a regular sequence of polytopes. Next, we introduce the notion of polytope syzygies.
Polytope Syzygies: A syzygy of a sequence of polytopes is an -tuple of elements in such that satisfies the following property:
every vertex in is shared by at least two elements in .
We say that the polytope syzygy is -dimensional if all the polytopes have dimension exactly .
This definition is inspired by the notion of tropical linear dependence studied by Jensen and Payne . A collection of tropical polynomials in -variables are said to be tropically dependent if there exist such that for every the minimum over is attained by at least two elements in . This definition is in the same spirit as the notion of tropical rank of a matrix studied in .
Extending this definition to the notion of syzygy of tropical polynomials, a tuple of tropical polynomials in -variables is a tropical syzygy if for every the minimum over is attained by at least two elements in . In the language of Newton polytopes, this translates to the notion of polytope syzygies. ∎
Every syzygy has a polytope associated to it. The -tuple is always a syzygy since, the corresponding polytope is and this is shared by every element in . A natural question in this context is the following:
Fix natural numbers , classify polytopes that are associated to a syzygy of polytopes where are -dimensional and are all distinct.
For , the answer is precisely those polytopes with vertices in that are decomposable into a Minkowski sum of two polytopes, both of which have dimension and similarly, arbitrary polytopes associated to Koszul syzygies are precisely of this form. The triangular prism shown in Figure 1 is an example of a polytope associated to a one-dimensional polytope syzygy of three polytopes. We are not aware of a classification of such polytopes. For instance, are there numerical invariants that completely characterize this? How does this property depend on the geometry of the polytope?
The set of all syzygies of form a semimodule of (see Proposition 5.1). We denote this semimodule by . The semimodule is not necessarily finitely generated in general, as the following example shows.
Consider the five polytopes where is the convex hull of and and , , and are the points , , and respectively. Consider the Minkowski sum of with any line segment joining and where is an integer vector with non-negative coordinates that is not a scalar multiple of . This is a parallelogram. Syzygies of are obtained by translating the other four polytopes (these are points) to the vertices of this parallelogram. If is a primitive vector, then the corresponding syzygy is a minimal generator of and hence, is not finitely generated.
Given an -tuple of polynomials, any syzygy between them specializes to a polytope syzygy via the Newton polytope map.
Let be an -tuple of polynomials in . Suppose that is a syzygy of , then is a polytope syzygy of .
In fact, a statement stronger than Proposition 1.8 holds: every lattice point in is shared by at least two elements in . The notion of polytope syzygy only captures the “convex part” of this property.
A general strategy for proving results about syzygies of polynomials is to study (polytope) syzygies of their Newton polytopes and keep track of which polytope syzygies lift.
The polytope associated to a polytope syzygy induces a natural equivalence between polytope syzygies. Two syzygies between the same collection of polytopes are said be equivalent if their associated polytope is the same and the set of coordinates where is the same for both. In the following, we present some more examples of polytope syzygies:
Koszul Syzygies: For a pair of polytopes , the pair is a syzygy with associated polytope . As in the case of commutative rings, we call this syzygy the Koszul syzygy of and . Similarly, for a collection of polytopes and for , the -tuple
is a syzygy with associated polytope .
In general, a collection of polytopes has syzygies other than the Koszul syzygies. Consider the triangular prism shown in Figure 1. Consider its three quadrilateral faces and let be the three line segments shown by the pointed curves in Figure 1. The triple , where and are the line segments corresponding to the edge of the quadrilateral adjacent to and respectively and is adjacent to in the remaining quadrilateral, is a syzygy. This is not a Koszul syzygy and is also not generated by Koszul syzygies.
Type of a Syzygy: Let be a syzygy of polytopes with associated polytope , then the syzygy is said to be of type , if is the minimum cardinality of a subset of such that .
Any Koszul syzygy is of type , while the syzygy in Figure 1 is of type . By definition, every syzygy of a collection of polytopes is of type at most .
Characterizing syzygies of a collection of polytopes is a guiding question in this context. In the case of commutative rings, we have the following characterization of syzygies of a regular sequence.
(Koszul Property)[19, Proposition 2] For elements in a commutative ring , let be the submodule of generated by the Koszul syzygies. The equality holds if and only if form a regular sequence.
Let be a sequence of polytopes. Let be the sub-semimodule of generated by the Koszul syzygies. Suppose that is a syzygy of polytopes , let be the polytope associated to it. Suppose that the syzygy is of type-I, then there is an index such that . The set of all indices for which is called the index set of the type-I syzygy. We show an analogous characterization of regular sequences of polytopes.
(Weak Koszul Property for Polytopes) Let be a sequence of polytopes in . Every type-I syzygy of is equivalent to a type-I syzygy in whose index set contains the index set of if and only if is a regular sequence.
On the other hand, the collection of polytopes in Figure 1 is an example of a regular sequence of polytopes that has a syzygy (for instance, the syzygy in Figure 1) that is not equivalent to any element in .
1.1 Motivation and Related Work
The direction of developing linear algebra and algebraic geometry over polytope semirings was suggested by Speyer and Sturmfels in 2009 . There is a substantial literature on linear algebra over the tropical semiring, see for example .
In this paper, we take first steps towards algebraic geometry over polytope semirings. We briefly describe previous work that served as impetus for us. One thread was the work of Bayer and Eisenbud on graph curves in 1991 . This paper was motivated by Green’s conjecture on syzygies of a smooth proper algebraic curve. In particular, Bayer and Eisenbud formulate a conjecture for graph curves analogous to Green’s conjecture on a smooth algebraic curve in its canonical embedding. They also proved their conjecture for graph curves where the underlying graph is planar. But, the general case is still open. However, as they point out in their paper, their conjecture does not imply Green’s conjecture. The reason is that the Clifford index of a graph curve of genus is too small (of the order ) compared to , the Clifford index of a general proper, smooth curve of genus . This is an obstacle to carrying out standard degeneration arguments.
Over the past ten years, there has been significant progress in tropical geometry that opens up the possibility of using tropical curves instead of graph curves. In particular, families of abstract tropical curves with Clifford index (defined in the sense of divisor theory of tropical curves) equal to that of a generic smooth curve have been constructed . Hence, this family of abstract tropical curves can substitute for graph curves provided that there is a notion of tropical (or polytope) syzygy that behaves “well” with respect to degeneration. A goal of this paper is to serve as a first step in this direction.
Polytope semirings (under the name polytope algebra) appeared in the book of Pachter and Sturmfels [15, Chapter 2] in the context of computational biology. Semiring theory, in particular idempotent semiring theory, has been treated in several books and articles, see for example . Linear algebra over semirings has been the focus of these works.
Recent works of MacPherson ,  introduce an analogue of integral closure over idempotent semirings and a notion of projective modules over polyhedral semirings. Our work is another step towards commutative algebra over semirings. Our results are over polytope semirings and do not seem to directly extend to arbitrary semirings. This is primarily because of our use of the Lebseque measure on the set of polytopes. Other contexts where polytope semirings appear are Litvinov , Connes [3, Page 19]. Litvinov  emphasizes a correspondence principle between classical analysis and idempotent analysis. According to this principle, every result in classical analysis has an idempotent analogue. Theorem 3.7 and Theorem 5.3 confirm a similar interplay between commutative rings and polytope semirings.
Another object related to polytope semirings is McMullen’s polytope algebra . The polytope algebra is the vector space generated by all symbols of the form where is convex polytope in along with the relations
where is a convex polytope and for every . Multiplication is given by .
Addition in the polytope algebra seems to be more “rigid” than its counterpart in the polytope semiring and hence, the concept of syzygies in the polytope semiring does not seem to have a corresponding object in the polytope algebra. Syzygies in the polytope algebra have been considered in  but we are not aware of a concrete relation between them and the polytope syzygies studied in this paper.
In a recent paper, Rowen  introduces the concept of negation map on a semiring to handle the problem of the absence of additive inverses in a semiring. He systematically uses the negation map to define (and sometimes recover) several notions of linear algebra over semirings, the most relevant to this paper is linear dependence. This raises the question of whether we can use Rowen’s negation map to define our notion of syzygy in the polytope semiring. At the time of writing this paper, we are not aware of a negation map on the polytope semiring that can recover our notion of polytope syzygy. A simpler question is whether the notion of tropical linear dependence due to Jensen and Payne  is an instance of linear dependence on the tropical semiring with a suitable negation map, we are also not aware of this.
Acknowledgements: We thank Matt Baker, Spencer Backman, Alex Fink, Christian Haase and Bernd Sturmfels for stimulating discussions on the topic of this paper. We thank the anonymous referee for several constructive suggestions.
2 Basics of Polytope Semirings
In this section, we document properties of polytope semirings that we employ throughout the rest of the paper. Let be an ideal in and consider a polynomial . Let be the image of under the Newton polytope map and . By a sub-semimodule of , we mean a subset of that is also a semigroup under and satisfies .
The set is a sub-semimodule of the polytope semiring .
Let and for . We have . Furthermore, for generic choices of , we have . Since, has infinite cardinality such generic choices of and exist. Hence, we conclude that is a semigroup under and satisfies . Thus, is a sub-semimodule of . ∎
2.1 Graded Sub-semimodules of the Polytope Semiring
We start by making the notion of a graded sub-semimodule of precise. Recall the grading on defined in the introduction.
A sub-semimodule of the polytope semiring is graded if it can be decomposed into a disjoint union of (possibly empty) semigroups for such that and generates as a semigroup under .
If is a -graded ideal of , then is graded as a sub-semimodule of with the -th graded piece where is the additive identity of .
2.2 Measures on Polytope Semirings
Let be a monotonic measure of i.e., a measure such that if , for instance the Lebseque measure on . Let . Proposition 2.4 states that a monotonic measure on is monotonic under linear combinations of elements of . This property serves as a “substitute” for the lack of additive inverse in several arguments, for instance in Proposition 2.5 and the proof of the Koszul property for polytopes.
Suppose that a polytope satisfies for a (possibly infinite) family , then for every . Hence, for any monotonic measure on we have for every .
In the following proposition, we use Proposition 2.4 to show the uniqueness of minimal generating sets of sub-semimodules in .
Every sub-semimodule of the polytope semiring has a unique (but not necessarily finite) minimal generating set. A graded sub-semimodule has a unique (but not necessarily finite but countable) graded minimal generating set.
Suppose that and are two distinct minimal generating sets of . Suppose that . Since is a generating set, we can write
for over all . Note that we can assume that is not a translate of any element in , otherwise this would contradict that is a minimal generating set. Let be the Lebsegue measure on . By the Brunn-Minkowski inequality, for all and by Proposition 2.4, for all . Furthermore, suppose that is the relative measure of , then we have for all . On the other hand, each for can also be written as an -linear combination of . Since for all , none of these -linear combinations involve . Combining these linear combinations with (1), we can write as an -linear combination of . This contradicts our assumption that is a minimal generating set of . For a graded sub-semimodule, note that every minimal generating set is graded to conclude the uniqueness of a graded minimal generating set. The countability of a minimal generating set follows by observing that each graded piece has only a finite number of minimal generators. ∎
Let be a -graded ideal of . For every , the graded minimal generating set for (as a semi-group over ) lifts to a generating set for the vector space . Hence, for every , the -th Newton-Hilbert coefficient of is at least its -th Hilbert coefficient.
Let be the minimal generating set of . Consider polynomials such that . We prove the statement by induction on the number of lattice points of where . If the number of lattice points of is one, then is a monomial and for some . Hence, is equal to for some non-zero . Assume that the statement is true for polynomials in whose Newton polytope contains lattice points. Consider a polynomial such that has lattice points then, . Consider any vertex , we know that is contained in some polytope , say. Furthermore, . Hence, there exists an element such that contains at most lattice points and hence, by the induction hypothesis . This implies that . Hence, is a generating set of and as a consequence, the -th Newton-Hilbert coefficient of is at least its -th Hilbert coefficient. ∎
Let be a -graded ideal of . The graded minimal generating set for (as a semimodule over ) lifts to a generating set of .
Let be the graded minimal generating set of . Consider polynomials such that . We prove the statement by induction on the number of lattice points of where . If the number of lattice points of is one then is a translate of for some . Hence, is a monomial multiplied by . Assume that the statement is true for polynomials in whose Newton polytope has lattice points. Consider a polynomial with lattice points then, with for all . Consider any vertex , we know that is contained in some summand , say. Furthermore, . Hence, there exists an element such that and contains at most lattice points (in particular, does not contain ) and by the induction hypothesis . This implies that . ∎
2.3 Minkowski Semigroup
The set of lattice polytopes with vertices in with Minkowski sum as the operation is a semigroup . We refer to this semigroup as the Minkowski semigroup. The Minkowski semigroup is commutative i.e., for every pair and cancellative i.e., implies that .
An irreducible polytope in the Minkowski semigroup is a polytope that cannot be written as a Minkowski sum of two polytopes in the Minkowski semigroup neither of which is a point. In particular, every point is a irreducible polytope.
Every polytope in the Minkowski semigroup can be written, up to rearrangement, as a product of irreducible polytopes. This product is not necessarily unique.
2.4 Equations over the Polytope Semiring
Fix polytopes and in . Consider the equation
where each is an element in . We ask the following questions:
Does this equation have a solution?
Are there finitely many solutions?
If yes, can we count them?
Let be the set of all solutions to this equation. The set forms a semigroup under coordinate-wise addition. The following lemma asserts that is a finite set.
The set is finite.
Let be a solution to Equation (2). We have . Since and are lattice polytopes, there are only a finitely many choices for the polytope . More precisely, if is a vertex of then the vertices of are a subset of lattice points of translated by . ∎
When , Lemma 2.9 allows us to define the canonical solution to the Equation (2) obtained by summing (in other words, taking convex hull) over all elements in . Furthermore, if we fix another polytope and consider the set of all solutions to Equation (2) such that is a Minkowski summand of each element . This set of solutions is also finite and when , the canonical solution with respect to is defined as the sum of all elements in .
Suppose that is a Minkowski summand of and let . Assume that , the canonical solution is equal to where is the term-wise Minkowski sum of the canonical solution to the equation with .
The Minkowski sum of any element of with is an element in . Furthermore, if and suppose that for each from to . We claim that is the canonical solution to . Otherwise, the canonical solution satisfies for every and hence . By the cancellative property of the Minkowski semigroup, for some and this contradicts the uniqueness of the canonical solution with respect to .
3 Newton-Hilbert Series
In this section, we prove Theorem 1.4 of the introduction i.e., the rationality of the Newton-Hilbert series of a graded sub-semimodule of the polytope semiring that satisfies a property analogous to Cohen-Macaulayness. We start by recalling proofs for rationality of the Hilbert series over the polynomial ring.
Suppose that is a finitely generated graded module over the polynomial ring. By the Hilbert syzygy theorem, has a finite minimal free resolution. The finite minimal free resolution of is used to express the Hilbert series of as an alternating sum of the Hilbert series of free modules in each homological degree; the rationality of the Hilbert series of free modules is a simple computation. To the best of our knowledge, there is no known analog of the Hilbert syzygy theorem over the polytope semiring. Hence, we do not know if this proof can be adapted to prove the rationality of the Newton-Hilbert series.
Instead, we take cue from a different proof of the rationality of Hilbert series of Cohen-Macaulay modules that goes via induction on the depth of the module. The base case is that of Artinian modules and is immediate. Suppose that is a regular element of a graded module , consider the short exact sequence
The additivity of Hilbert series in a short exact sequence and the fact that the depth of is one less than the depth of then implies the rationality of the Hilbert series of .
In order to adapt this proof for the polytope semiring, we define a notion of regular sequence on a sub-semimodule . More precisely, we provide a condition for the sequence of coordinate points (Newton polytopes of variables ) to be regular.
Let be the sub-semigroup of containing as a Minkowski summand and be the sub-semigroup of not containing as a Minkowski summand. Note that is the disjoint union of and . In the following, we define the notion of a coordinate point to be a regular element on a sub-semimodule of .
A coordinate point is regular on a graded sub-semimodule of if and only if for all we have:
The following remarks clarify and motivate Definition 3.1.
The containment always holds and its converse is the non-trivial condition imposed by Definition 3.1.
Definition 3.1 is motivated by its analogue in the commutative ring setting: Let be a graded ideal of the polynomial ring . A homogenous element of degree one in the graded polynomial ring is a regular element (i.e., not a zero divisor) on if and only if for every where consists of all elements of degree in that divide .
Note the difference between the notion of regular sequence in the introduction and Definition 3.1: in the introduction, we gave a condition for an arbitrary (finite) sequence of polytopes to be a regular sequence on the polytope semiring whereas Definition 3.1 is a condition for a coordinate point to be regular on an arbitrary sub-semimodule of the polytope semiring. According to the following proposition, the two definitions agree when they are both meaningful.
Let be a regular sequence of polytopes on the polytope semiring such that is a graded sub-semimodule of . The coordinate point is regular (in the sense of Definition 3.1) on the sub-semimodule generated by if and only if is a regular sequence on .
If is regular on , then suppose that . Since is a graded sub-semimodule of we know that where each is contained in a graded piece of . Furthermore, each contains as a Minkowski summand. Hence, there exists a such that . This implies that showing that is a regular sequence.
Conversely, suppose that is regular, then if then . Hence, for every . ∎
We define the notion of regularity for a sequence of coordinate points on an arbitrary sub-semimodule of the polytope semiring.
Let be a sub-semimodule of the polytope semiring. A sequence of coordinate points is called a regular sequence on if the following conditions are satisfied:
The coordinate point is regular on and is regular in for where is the hyperplane of points with -th coordinate zero.
Let . For each and for from , we have .
Definition 3.3 is a condition for a sequence of coordinate points to be regular on an arbitrary graded sub-semimodule of the polytope semiring whereas the notion of regular sequence in the introduction is for an arbitrary sequence of polytopes to be regular on the polytope semiring. Both these definitions can be applied to a sequence of distinct coordinate points on the polytope semiring. A sequence of distinct coordinate points is a regular sequence according to both Definition 3.3 and the definition of regular sequences in the introduction.
The second condition in Definition 3.3 is motivated by its counterpart in the case of the polynomial ring: Let be a graded ideal in the polynomial ring . Suppose that is a regular sequence of . Consider the vector space spanned by elements in obtained by plugging in . Let be its subspace spanned by elements that are not divisible by . For every from two to and for every , the rank of is equal to the rank of . ∎
A sub-semimodule of is called Artinian if there exists a sufficiently large such that for all . Furthermore, it is called Cohen-Macaulay if there exists a regular sequence of coordinate points such that is Artinian over the copy of the polytope semiring generated by the coordinates in . The depth of is the smallest such integer .
In particular, if has depth zero then it is Artinian.
(Rationality of Newton-Hilbert Series) Let be a Cohen-Maculay graded sub-semimodule of , then its Newton-Hilbert series is a rational function.
We apply induction on the depth of . If is Cohen-Macaulay of depth zero then it is Artinian. Hence, its Newton-Hilbert series is rational. Assume that the Newton-Hilbert series of every Cohen-Macaulay graded sub-semimodule of depth at most is rational. Consider a Cohen-Macaulay graded sub-semimodule of depth . Suppose that is a regular sequence on .
Consider , the -th graded piece of . Decompose into and .
Hence, . Since is a regular element on , we have the following two properties:
This induces the following recurrence on the Newton-Hilbert series:
Hence, . Note that is a sub-semimodule of the copy of the polytope semiring generated by the last coordinates and has depth precisely . By the induction hypothesis, it has a rational Newton-Hilbert series. This implies that is also rational. ∎
Without the assumption of Cohen-Macaulayness, we are not able to show the rationality of the Newton-Hilbert series. The main difficulty is that the induction parameter depth is not apparent in the general case. In a recent work, Sam and Snowden  develop the concept of combinatorial categories and prove rationality results of the Hilbert series for representation of such categories. We are not aware of a concrete connection between polytope semirings and combinatorial categories. This may be a useful tool to proving rationality results of Newton-Hilbert series.
Remark also that regular elements on do not necessarily specialize to regular elements on . For instance, consider the toric ideal of . The variable is a regular element (since it is not a zero divisor) on . But is not a regular element of . For instance, for the second condition in the definition of regular sequence is violated: whereas .
The Newton-Hilbert series for the polytope semiring agrees with the graded polynomial ring in -variables. It is given by the rational function .
For any monomial ideal of , the Newton-Hilbert series of coincides with its Hilbert series .
In the following, we present an example of a family of Cohen-Macaulay sub-semimodules of the polytope semiring , each of depth one. Pick an natural number . Let be the set of all points in whose first coordinate is at least one and be its complement in . For each point in , consider the polytope given by the convex hull of .
Let be the sub-semimodule generated by over all . By construction, is Artinian as a sub-semimodule of the copy of corresponding to the semiring generated by the last two coordinates. Furthermore, is Cohen-Macaulay of depth one. The same computation as in the proof of Theorem 3.7 shows that its Newton-Hilbert series is . Using a similar approach, we can construct Cohen-Macaulay sub-semimodules of arbitrary depth.
4 Regular Sequence of Polytopes
We construct examples of regular (and non-regular) sequences of polytopes. We start with the case of two polytopes.
If two polytopes share a non-trivial summand i.e., a Minkowski summand that is not a point then they do not form a regular sequence
The converse, however is not true. For instance in , let be the rectangle with the four vertices and be the triangle with vertices