Heights of Degeneracy Loci

Given an f × e matrix ϕ of homogeneous polynomials in several variables with coefficients in a field, the bit size of a Gröbner basis for the ideal defining the locus of points where ϕ fails to attain some given rank k is closely related to the height of this locus. Using the arithmetic intersection theory of Gillet and Soulé together with the effective arithmetic Schubert calculus of Maillot, we compute this height and relate it to the bit complexity of solving certain polynomial optimization problems.