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arXiv · 2111.06993

Covering Weight-Determined Sets of Uniform Grids by Polynomials

S. Venkitesh
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Abstract

We determine the minimum degree of a nonzero polynomial covering "weight-determined" subsets of a uniform grid over a field of characteristic zero. Since such subsets are completely determined by the grid dimensions and the set of weights taken by the points, we see that the minimum degree is also determined by the set of weights. In fact, the minimum degree can be algorithmically computed in time linear in the grid dimensions. Previously, such a result and algorithm was known only for uniform grids that are "strictly unimodal" (Venkitesh, E-JC 2022), which implies the grid cannot be "lopsided". Our result holds in full generality for all uniform grids. Our central tools are the classical algebraic objects called the "finite-degree Zariski closure" and the "affine Hilbert function". Our technical contribution is a combinatorial characterization of both these objects for all weight-determined subsets of a uniform grid. The minimal degree of our covering problem is precisely the threshold degree at which the behavior of the Zariski closure changes sharply.

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S. Venkitesh. 2021-11-12. Covering Weight-Determined Sets of Uniform Grids by Polynomials. https://arxiv.org/abs/2111.06993

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