On arithmetical bounds of Chow-forms

Computing Bounds from Arithmetical Proofs

We explore the role of the function a + 2, and its generalizations to higher number classes, in providing complexity bounds for the provably computable functions across a broad spectrum of theories, based on a “predicative” induction scheme and ranging in strength between polynomial-time arithmetic and Π11-CA0. The resulting “fast-growing” sub-recursive hierarchy forges a direct link between pr...

The Arithmetical Invariants of Quadratic Forms

Computing Chow Forms and Some Applications

We prove the existence of an algorithm that, from a finite set of polynomials defining an algebraic projective variety, computes the Chow form of its equidimensional component of the greatest dimension. Applying this algorithm, a finite set of polynomials defining the equidimensional component of the greatest dimension of an algebraic (projective or affine) variety can be computed. The complexi...

The Zarankiewicz Problem via Chow Forms

The well-known Zarankiewicz problem [Za] is to determine the least positive integer Z(m,n, r, s) such that each m × n 0-1 matrix containing Z(m,n, r, s) ones has an r × s submatrix consisting entirely of ones. In graph-theoretic language, this is equivalent to finding the least positive integer Z(m, n, r, s) such that each bipartite graph on m black vertices and n white vertices with Z(m,n, r, ...

Resultants and Chow Forms via Exterior Syzygies

Let W be a vector space of dimension n+ 1 over a field K. The Chow divisor of a k-dimensional variety X in P = P(W ) is the hypersurface, in the Grassmannian Gk+1 of planes of codimension k+1 in P, whose points (over the algebraic closure of K) are the planes that meet X . The Chow form of X is the defining equation of the Chow divisor. For example, the resultant of k+ 1 forms of degree e in k ...