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Source. Definition 4.1 and the remarks after it, p. 15, 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.

Statement

Definition 4.1 (p. 15). For a lattice $\mathcal L\subseteq\mathbf Z^n$ with Graver basis $\operatorname{Gr}_{\mathcal L}$ (Definition 2.5, p. 4, on the Theorem 2.8 page), the product ideal is the monomial ideal

$$P_{\mathcal L}=\left\langle x^ux^v:u-v\in\operatorname{Gr}_{\mathcal L}\right\rangle .$$

Here $u,v\in\mathbf N^n$. Every such $u,v$ dominate the positive and negative parts $g^+,g^-$ of $g=u-v$, so the same ideal is generated by the monomials $x^{g^++g^-}$, $g\in\operatorname{Gr}_{\mathcal L}$.

The paper records (p. 15) that $P_{\mathcal L}\subseteq V_{\mathcal L}$, because every generic initial ideal $\operatorname{in}_\omega(I_{\mathcal L})$ contains $x^u$ or $x^v$ whenever $u-v\in\mathcal L$, and that the containment can be strict: for $\mathcal L=\ker(A)\cap\mathbf Z^3$ with $A=[3\ 4\ 5]$, $P_{\mathcal L}=\langle ab^2c,a^2bc^2,a^3bc,a^4b^3,a^5c^3,b^5c^4\rangle$ is strictly contained in $V_{\mathcal L}=\langle ab^2c,a^2bc,a^4b^3,a^5c^3,b^5c^4\rangle$.

Read depth. Claims checked: the definition, the containment and its one-line justification, and the example were read on p. 15; the example's ideals were not recomputed.

Dependencies

Theorem 2.3 for the containment; Definition 2.5 for the Graver basis.

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 derives from this definition that, for the lattice $\mathcal L_X$ of integer relations among the elements of a finite set $X\subset\mathbf R$, a subset $S$ is dissociated if and only if the squarefree monomial $x_S$ is not in $P_{\mathcal L_X}$, so the largest dissociated subset has the largest degree of a squarefree standard monomial of $P_{\mathcal L_X}$. That is a reformulation; the paper gives no bound on that degree, and neither the problem's lower bound nor a counterexample follows from it.