Papers › Ideals of Spaces of Degenerate Matrices
Ideals of Spaces of Degenerate Matrices
Julian Vill, Mateusz Michałek, Alexander Taveira Blomenhofer
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The variety Sing_(n, m) consists of all tuples X = (X₁,…, Xₘ) of n×n matrices such that every linear combination of X₁,…, Xₘ is singular. Equivalently, X∈Sing_(n,m) if and only if det(λ₁ X₁ + …+ λₘ Xₘ) = 0 for all λ₁,…, λₘ∈ℚ. Makam and Wigderson 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(λ₁ X₁ + …+ λₘ Xₘ). We answer this question in the negative by determining the vanishing ideal of Sing_(2, 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 Sing_(n, m) that are not in the ideal generated by polynomials of type det(λ₁ X₁ + …+ λₘ Xₘ). Our methods are based on classical results about Fano schemes, representation theory and Gr\"obner bases.
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