Vanishing ideals and codes on toric varieties

Abstract: Motivated by applications to the theory of error-correcting codes, we give an algorithmic method for computing a generating set for the ideal generated by $\beta$-graded polynomials vanishing on a subset of a simplicial complete toric variety $X$ over a finite field $\mathbb{F}_q$, parameterized by rational functions, where $\beta$ is a $d\times r$ matrix whose columns generate a subsemigroup $\mathbb{N}\beta$ of $\mathbb{N}^d$. We also give a method for computing the vanishing ideal of the set of $\mathbb{F}_q$-rational points of $X$. We talk about some of its algebraic invariants related to basic parameters of the corresponding evaluation code. When $\beta=[w_1 \cdots w_r]$ is a row matrix corresponding to a numerical semigroup $\mathbb{N}\beta=\langle w_1,\dots,w_r \rangle$, $X$ is a weighted projective space and generators of its vanishing ideal is related to the generators of the defining (toric) ideals of some numerical semigroup rings corresponding to semigroups generated by subsets of $\{w_1,\dots,w_r\}$.

algebraic geometry

Audience: researchers in the discipline

ODTU-Bilkent Algebraic Geometry Seminars