Let $K$ be a finite field and $X$ be a complete simplicial toric variety over $K$ with split torus $T_X\cong (K^*)^n$. We give an algorithm relying on elimination theory for finding generators of the vanishing ideal of a subgroup $Y_Q$ parameterized by a matrix $Q$ which can be used to study algebraic geometric codes arising from $Y_Q$. We give a method to compute the lattice $L$ whose ideal $I_L$ is exactly $I(Y_Q)$ under a mild condition. As applications, we give precise descriptions for the lattices corresponding to some special subgroups. We also prove a Nullstellensatz type theorem valid over finite fields, and share Macaulay2* codes for our algorithms.
Evaluation code, toric variety, multigraded Hilbert function, vanishing ideal, parameterized code, lattice ideal
ÖZKAN, ESMA BARAN
"Vanishing ideals of parameterized subgroups in a toric variety,"
Turkish Journal of Mathematics: Vol. 46:
6, Article 7.
Available at: https://journals.tubitak.gov.tr/math/vol46/iss6/7