The variety Sing(n,m) consists of all tuples X=(X-1,..., X-m) of n x n matrices such that every linear combination of X-1,..., X-m is singular. Equivalently, X is an element of Sing(n,m) if and only if det(lambda X-1(1)+...+lambda X-m(m)) = 0 for all lambda(1),...,lambda(m) is an element of Q. Makam and Wigderson [12] asked whether the ideal generated by these equations is always radical, that is, if any polynomial identity that is valid on Sing(n,m) lies in the ideal generated by the polynomials det(lambda X-1(1)+...+lambda X-m(m)). We answer this question in the negative by determining the vanishing ideal of Sing(2,m) for all m is an element of N. Our results exhibit that there are additional equations arising from the tensor structure of X. More generally, for any nand m >= n(2) - n + 1, we prove there are equations vanishing on Sing(n,m) that are not in the ideal generated by polynomials of type det(lambda X-1(1)+...+lambda X-m(m)). Our methods are based on classical results about Fano schemes, representation theory and Grobner bases. (C) 2022 Elsevier Inc. All rights reserved.
Multilinear algebraTensor calculusComputational aspects of algebraic geometry
Vill, Julian、Michalek, Mateusz、Blomenhofer, Alexander Taveira
NSTL
Elsevier
