New advances in the study of Gröbner bases and the number of latin squares related to autotopisms

Abstract

Gröbner bases has been used in [4] to describe an algorithm that allows one to obtain the number of Latin squares of order up to 7 having a given isotopism in their autotopism group. In order to improve the time of computation of this algorithm, we study in this poster a possible combination between Gröbner bases and some combinatorial tools. Specifically, we add to the ideal of polynomials defining a Latin square L, some polynomials related to the permutations of rows, columns and symbols corresponding to the given autotopism of L. Using this method we could obtain the number of some Latin squares of order 8 having an isotopism in their autotpism group.