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Source. Proposition 2.1, p. 3, with Definition 1.1 (p. 1), of Serkan Hoşten and Diane Maclagan, The vertex ideal of a lattice, arXiv:math/0012197v1 (2000), published in Adv. in Appl. Math. 29 (2002), 521--538, as identified on the source card. Page numbers are those of the arXiv print.

Setting

Let $\mathcal L$ be a lattice in $\mathbf Z^n$ with $\dim(\mathcal L)=m$ (Definition 1.1, p. 1). For $u\in\mathbf N^n$ the fiber of $u$ is

$$P_u=\operatorname{conv}\{v\in\mathbf N^n:u-v\in\mathcal L\},$$

and $P_v=P_u$ whenever $v\in P_u$. Each fiber is a rational polyhedron (Theorem 16.1 of Schrijver's Theory of Linear and Integer Programming, cited on p. 1), so it has finitely many vertices $\operatorname{Vert}(P_u)$. Let $S=k[x_1,\ldots,x_n]$ and let $e_i$ be the $i$-th unit vector.

Statement

Proposition 2.1 (p. 3). Let $P_u$ be a fiber of $\mathcal L$. If $v$ is a vertex of $P_u$ and $v_i>0$, then $v-e_i$ is a vertex of its own fiber (the print writes that fiber as $P_{u-e_i}$). Equivalently, there is a monomial ideal $V_{\mathcal L}\subseteq S$ such that $x^v\notin V_{\mathcal L}$ if and only if $v\in\operatorname{Vert}(P_u)$ for a fiber $P_u$ of $\mathcal L$.

The paper calls $V_{\mathcal L}$ the vertex ideal of $\mathcal L$ (p. 1): its standard monomials are exactly the monomials whose exponents are vertices of their fibers, and the union of all $\operatorname{Vert}(P_u)$, $u\in\mathbf N^n$, is an order ideal of $\mathbf N^n$.

Read depth. Claims checked: the statement and Definition 1.1 were read clause by clause on pp. 1 and 3, with the short proof on p. 3.

Proof pointer

Page 3. If $v-e_i$ were a convex combination of other points of its fiber, adding $e_i$ to each of them would write $v$ as a convex combination of other points of $P_u$.

Dependencies

Definition 1.1 of the same paper; finiteness of the vertex set of a rational polyhedron, cited from Schrijver.

Bears on

• Problem 963: background only. The paper does not mention dissociated sets, subset sums or the problem. The source card's section on E963 uses this proposition to read $x_S\notin V_{\mathcal L_X}$, for the lattice $\mathcal L_X$ of integer relations among the elements of $X$, as the statement that the incidence vector of $S$ is a vertex of its fiber, which with Definition 4.1 implies that $S$ is dissociated; it gives no bound on the size of such $S$.