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Degree and algebraic properties of lattice and matrix ideals. (English) Zbl 1334.13017
Let $$\mathcal{L}$$ be a lattice in $$\mathbb{Z}^s$$ of rank $$s-1$$, generated by $$\alpha_1,\ldots,\alpha_m$$ and let $$L$$ be the $$s\times m$$ matrix with column vectors $$\alpha_1,\ldots,\alpha_m$$. Each $$\alpha_i$$ can be written uniquely as $$\alpha_i=\alpha_i^+-\alpha_i^-$$, where $$\alpha_i^+,\alpha_i^-\in \mathbb{N}^s$$. The matrix ideal of $$L$$, denoted by $$I(L)$$, is generated by all the binomials $$t^{\alpha_i^+}-t^{\alpha_i^-}$$, for all $$i=1,\ldots,m$$.
Consider a polynomial ring $$\mathbb{K}[x_1]$$ in one variable, where $$\mathbb{K}$$ is a field containing the $$\gamma_{s-1}$$-th roots of unity with $$\text{char}(\mathbb{K})=0$$ or $$\text{char}(\mathbb{K})=p$$, where $$p$$ is a prime number with $$p\nmid\gamma_{s-1}$$. By $$\Lambda_i$$ is denoted the set of $$\gamma_i$$-th roots of unity in $$\mathbb{K}$$ and let $$\Lambda=\prod_{i=1}^{s-1}\Lambda_i$$. For any $$\lambda=\left(\lambda_1,\ldots,\lambda_{s-1}\right)\in\Lambda$$, there is an homomorphism of $$\mathbb{K}$$-algebras: $\phi_{\lambda}: S\longrightarrow\mathbb{K}[x_1],\;t_i\longmapsto \lambda_1^{p_{1,i}}\cdots\lambda_{s-1}^{p_{s-1,i}},\;i=1,\ldots,s.$ The kernel of $$\phi_{\lambda}$$, denoted by $$\alpha_{\lambda}$$, is a prime ideal of $$S$$ and of height $$s-1$$.
The authors in their first main result (Theorem 3.3) prove for a graded lattice ideal $$I(\mathcal{L})$$ of dimension one, that $$I(\mathcal{L})=\bigcap_{\lambda\in\Lambda}\alpha_{\lambda}$$ is the minimal primary decomposition of the ideal $$I(\mathcal{L})$$ into exactly $$\gamma$$ primary components. By this result, they are able to characterize when the graded binomial ideals, satisfying the vanishing condition $$V(I,t_i)=\{0\}$$, for all $$i$$, are lattice ideals (Proposition 5.3). Moreover, they apply this proposition to give a structure of graded matrix ideals $$I$$ (Proposition 5.7), in terms of the notion of unmixed ideals; (ideals whose all associated primes have the same height) and the meaning of the intersection of the isolated primary components of $$I$$, which is denoted by $$\text{Hull}(I)$$.
In their next main result, the authors compute the degree of any lattice ideal (Theorem 4.6). More explicitly, they prove that if $$\mathcal{L}\subset\mathbb{Z}^s$$ is a lattice of rank $$r$$, then the following hold:
(a)
if $$r=s$$, then $$\deg\left(S/I\left(\mathcal{L}\right)\right)=\mid\mathbb{Z}^s/\mathcal{L}\mid$$.
(b)
if $$r<s$$, there is an integer matrix $$A$$ of size $$(s-r)\times s$$ with $$\text{rank}(A)=s-r$$ such that we have the containment of rank $$r$$ lattices $$\mathcal{L}\subset\ker_{\mathbb{Z}(A)}$$ with equality, if and only if $$\mathbb{Z}^s/\mathcal{L}$$ is torsion free.
(c)
if $$r<s$$ and $$v_1,\ldots,v_s$$ are the columns of $$A$$, then $\deg\left(S/I(\mathcal{L})\right)=\dfrac{\mid T(\mathbb{Z}^s/\mathcal{L})\mid(s-r)!\text{vol}(\text{conv}(0,v_1,\ldots,v_s))}{\mid T(\mathbb{Z}^{s-r}/ \langle v_1,\ldots,v_s\rangle) \mid},$ where by $$\text{vol}(P)$$ is denoted the relative volume of a lattice polytope $$P$$ and by $$\text{conv}(A)$$ the convex hull of a set $$A$$. It is remarked here, that for the well known result (b), the authors give a different proof, from which there is a way to compute the degree of an arbitrary lattice ideal. Also, they present some applications of the above theorem, to compute the corresponding degrees of certain families of lattice ideals.
In the last three sections of their article, the authors study certain classes of integer matrices and their corresponding ideals. They define the pure binomial matrix (PB matrix, for short), the positive pure binomial matrix (PPB matrix), the critical binomial matrix (CB matrix), the positive critical binomial matrix (PCB matrix), the generalized critical binomial matrix (GCB matrix) and the generalized positive critical binomial matrix (GPCB matrix). More explicitly, in section 6, applying all the above results, they prove some results for $$\text{GPCB}$$ matrices (resp. ideals), either generalizing or extending some results of L. O’Carroll and F. Planas-Vilanova [J. Algebra 373, 392–413 (2013; Zbl 1274.13002)] that hold for $$\text{PCB}$$ matrices (resp. ideals). In section 7, they study how their results apply to the family of binomial ideals arising from Laplacian matrices. The Laplacian matrix is an example of a CB matrix. In the last section, the authors describe the structure of lattice ideals of dimension 1 in three variables. In more details, they prove in the main result of this section that if $$S=\mathbb{K}[x_1,x_2,x_3]$$ is a polynomial ring and $$I$$ an ideal of $$S$$, then $$I$$ is a homogeneous lattice ideal of dimension one, if and only if it is a CB ideal. Finally, they add a new condition for a GPCB ideal to be a lattice ideal, like the Propositions 5.3 and 5.7, that where described above.

##### MSC:
 13F20 Polynomial rings and ideals; rings of integer-valued polynomials 13A15 Ideals and multiplicative ideal theory in commutative rings 13H15 Multiplicity theory and related topics 13P05 Polynomials, factorization in commutative rings 05E40 Combinatorial aspects of commutative algebra
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