bibkey: sullivant2006toricfiber authors: Seth Sullivant year: 2006 title: Toric Fiber Products doi: 10.48550/arXiv.math/0602052 url: https://arxiv.org/abs/math/0602052v1 claim: “Example 1.2 identifies the kernel of the Segre monomial map with the ideal of two-by-two minors; the general generating-set results require the stated multigrading hypotheses.” strata_touched: [] license: citation-only triage: anchor
The square minor and its restricted algebraic use
The primary version is arXiv:math/0602052v1. Definition 1.1, PDF p. 2, defines a toric fiber product as the kernel of the specified multigraded map. Example 1.2 on the same page takes and identifies the kernel as the ideal generated by all two-by-two minors. For a two-by-two table its single generator is
The pyramid reference, §十二 uses that square relation as a classical algebraic ingredient. It gives its own explicit monomial map, including the apex variable, and proves the kernel and the unique nonnegative normal form by the integer occupancy fiber. The subtraction of weak-composition counts and the resulting generating function are shown in that consumer; they are not quoted as formulas from this paper.
The introduction and §2 require the indicated linearly independent degree set for the general generating-set and Gröbner-basis results. No unrestricted gluing theorem is imported into the five-mode system. An integer circuit or an algebraic fiber product supplies no probability independence, chronological Markov assumption, native instrument or recovery of arbitrary source histories.