Binary determinantal complexity

Binary Determinantal Complexity

We prove that for writing the 3 by 3 permanent polynomial as a determinant of a matrix consisting only of zeros, ones, and variables as entries, a 7 by 7 matrix is required. Our proof is computer based and uses the enumeration of bipartite graphs. Furthermore, we analyze sequences of polynomials that are determinants of polynomially sized matrices consisting only of zeros, ones, and variables. ...

Determinantal complexity

Bi-polynomial rank and determinantal complexity

The permanent vs. determinant problem is one of the most important problems in theoretical computer science, and is the main target of geometric complexity theory proposed by Mulmuley and Sohoni. The current best lower bound for the determinantal complexity of the d by d permanent polynomial is d/2, due to Mignon and Ressayre in 2004. Inspired by their proof method, we introduce a natural rank ...

Depth-4 Lower Bounds, Determinantal Complexity: A Unified Approach

Tavenas has recently proved that any nO(1)-variate and degree n polynomial in VP can be computed by a depth-4 ΣΠ[O( p n)]ΣΠ[ p n] circuit of size 2O( p n log n) [Tav13]. So to prove VP 6= VNP, it is sufficient to show that an explicit polynomial ∈ VNP of degree n requires 2ω( p n log n) size depth-4 circuits. Soon after Tavenas’s result, for two different explicit polynomials, depth-4 circuit s...

A lower bound for the determinantal complexity of a hypersurface

We prove that the determinantal complexity of a hypersurface of degree d > 2 is bounded below by one more than the codimension of the singular locus, provided that this codimension is at least 5. As a result, we obtain that the determinantal complexity of the 3×3 permanent is 7. We also prove that for n > 3, there is no nonsingular hypersurface in Pn of degree d that has an expression as a dete...