Tim field of computational complexity for concrete, practical combinatorial problems has developed in a remarkably smt~,.th fashi.rt One can point to several t'catures of the theory of polyoowaal-time computability which make it especially well-behaved, including: (1) the modelling of feasible computing by polynomial-time complexity is well-supported by the fact that Mmost all known polynomial-...

The coloured Tutte polynomial by Bollobás and Riordan is, as a generalisation of the Tutte polynomial, the most general graph polynomial for coloured graphs that satisfies certain contractiondeletion identities. Jaeger, Vertigan, and Welsh showed that the classical Tutte polynomial is #P-hard to evaluate almost everywhere by establishing reductions along curves and lines. We establish a similar...

We prove lower bounds of order n logn for both the problem of multiplying polynomials of degree n, and of dividing polynomials with remainder, in the model of bounded coefficient arithmetic circuits over the complex numbers. These lower bounds are optimal up to order of magnitude. The proof uses a recent idea of R. Raz [Proc. 34th STOC 2002] proposed for matrix multiplication. It reduces the li...

We prove polynomial time complexity for a now widely used factorization algorithm for polynomials over the rationals. Our approach also yields polynomial time complexity results for bivariate polynomials over a finite field.

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 ...

A polynomial ideal membership problem is a (w+1)-tuple P = (f; g 1 ; g 2 ; : : : ; g w) where f and the g i are multivariate polynomials over some ring, and the problem is to determine whether f is in the ideal generated by the g i. For polynomials over the integers or rationals, it is known that this problem is exponential space complete. We discuss complexity results known for a number of pro...

We present three algorithms in this paper: the first algorithm solves zero-dimensional parametric homogeneous polynomial systems with single exponential time in the number n of the unknowns, it decomposes the parameters space into a finite number of constructible sets and computes the finite number of solutions by parametric rational representations uniformly in each constructible set. The seco...

A polynomial complexity algorithm is designed which tests whether a point belongs to a given tropical linear variety.

The representation of functions as low-degree polyno-mials over various rings has provided many insights in the theory of small-depth circuits. We survey some of the closure properties, upper bounds, and lower bounds obtained via this approach.

In this paper, we survey some of our new results on the complexity of a number of problems related to polynomial ideals. We consider multivariate polynomials over some ring, like the integers or the rationals. For instance, a polynomial ideal membership problem is a (w + 1)-tuple P = ( f, g1, g2, . . . , gw) where f and the gi are multivariate polynomials, and the problem is to determine whethe...