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Source. Proposition 4.2, p. 16, 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

A $d\times n$ matrix is unimodular if all its maximal $d\times d$ minors have the same absolute value (p. 16). $P_{\mathcal L}$ is the product ideal, and $I_{\Delta(\mathcal M(\mathcal L))}$ the Stanley–Reisner ideal of the matroid complex of Corollary 2.12.

Statement

Proposition 4.2 (p. 16). If $\mathcal L=\ker(A)\cap\mathbf Z^n$ where $A$ is a unimodular matrix, then $P_{\mathcal L}=V_{\mathcal L}$, and both equal $I_{\Delta(\mathcal M(\mathcal L))}$.

Read depth. Claims checked: the statement was read clause by clause on p. 16, with the proof.

Proof pointer

Page 16. For unimodular $A$ every initial ideal of $I_{\mathcal L}$ is squarefree (Sturmfels, Gröbner Bases and Convex Polytopes, Corollary 8.9), so $V_{\mathcal L}$ is radical and equals the matroid ideal by Corollary 2.12; the Graver basis consists of the circuits (ibid., Proposition 8.11), whose products of variables are exactly the matroid ideal's minimal generators.

Dependencies

Corollary 2.12; B. Sturmfels, Gröbner Bases and Convex Polytopes, American Mathematical Society, 1996, Corollary 8.9 and Proposition 8.11.

Bears on

No problem page of this corpus.