Abstract
The variety Singn,m consists of all tuples X=(X1,…,Xm) of n×n matrices such that every linear combination of X1,…,Xm is singular. Equivalently, X∈Singn,m if and only if det(λ1X1+…+λmXm)=0 for all λ1,…,λm∈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 Singn,m lies in the ideal generated by the polynomials det(λ1X1+…+λmXm). We answer this question in the negative by determining the vanishing ideal of Sing2,m for all m∈N. Our results exhibit that there are additional equations arising from the tensor structure of X. More generally, for any n and m≥n2−n+1, we prove there are equations vanishing on Singn,m that are not in the ideal generated by polynomials of type det(λ1X1+…+λmXm). Our methods are based on classical results about Fano schemes, representation theory and Gröbner bases.
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