Ideals, determinants, and straightening: proving and using lower bounds for polynomial ideals

No short polynomials vanish on bounded rank matrices

“…For another link of our work to polynomial identity testing, we refer to [1], where it is shown that any non‐zero element $$f$$ in the ideal generated by $$(r+1) \times (r+1)$$‐minors can be used as an oracle in the construction of a small circuit that approximately computes the $$s \times s$$‐determinant, for $$s=\Theta (r^{1/3})$$. This can be understood as expressing that such a polynomial has high border complexity , a different measure of complexity than the number of terms considered in this paper.…”

On Matrix Multiplication and Polynomial Identity Testing

“…Using an observation due to Baur and Strassen [BS83, Corollary 6], a lower bound on the border rank of matrix multiplication lifts to a lower bound on the border multiplicative complexity of the polynomial tr(XY Z), where X, Y , and Z are n × n matrices. Results of Andrews and Forbes [AF22] allow us to further lift this lower bound to the ideal I r where r is the size of the smallest algebraic branching program computing tr(XY Z). Because tr(XY Z) can be computed by an algebraic branching program with O(n 2 ) vertices, we obtain a lower bound of Ω(R( √ r)) on the multiplicative complexity of I r .…”

“…We denote by I det n,m,r ⊆ F[X] the ideal of F[X] generated by the r × r minors of X. We make use the following proposition of Andrews and Forbes [AF22], which reduces the task of proving lower bounds on all polynomials in I det n,m,r to the task of proving lower bounds on products of minors. We note that the polynomial (K σ |K σ )(X) appearing in the statement of [AF22, Proposition 3.5] is exactly the same as the product of determinants that appears in the proposition below.…”

“…) be the border rank of n × n × n matrix multiplication, our goal will be to prove a lower bound of order R(r) on the border multiplicative complexity of the ideal I det n,m,r . To do this, we make use of tools recently developed by Andrews and Forbes [AF22] to prove lower bounds on the complexity of polynomials in this ideal.…”