We introduce the notion of N-reduced dynamical cocycles and use these objects to define enhancements of the rack counting invariant for classical and virtual knots and links. We provide examples to show that the new invariants are not determined by the rack counting invariant, the Jones polynomial or the generalized Alexander polynomial.

This paper provides both positive and negative results for counting solutions to systems of polynomial equations over a finite field. The general idea is to try to reduce the problem to counting solutions to a single polynomial, where the task is easier. In both cases, simple methods are utilized that we expect will have wider applicability (far beyond algebra). First, we give an efficient dete...

We characterize the computational complexity of counting the exact number of satisfying assignments in (XOR; AND)-formulas in their RSE-representation (i.e., equivalently, polyno-mials in GF 2]]x 1 ; : : : ; x n ]). This problem refrained for some time eeords to nd a polynomial time solution and the eeorts to prove the problem to be #P-complete. Both main results can be generalized to the arbit...

We describe fully polynomial randomized approximation schemes for the problems of determining the number of Hamilton paths and cycles in an n-vertex graph with minimum degree (g + e)n, for any fixed e > 0. We show that the exact counting problems are #P-complete. We also describe fully polynomial randomized approximation schemes for counting paths and cycles of all sizes in such graphs.

The gcd-sum is an arithmetic function defined as the sum of the gcd’s of the first n integers with n : g(n) = ∑n i=1(i, n). The function arises in deriving asymptotic estimates for a lattice point counting problem. The function is multiplicative, and has polynomial growth. Its Dirichlet series has a compact representation in terms of the Riemann zeta function. Asymptotic forms for values of par...

To see the issues that arise when studying incidence problems in higher dimensions, we consider one of the simplest cases: Incidences between m points and n planes in R. To see that this problem is not interesting, we consider the following point-plane configuration. Let l ⊂ R be a line, let P be a set of m points on l, and let H be a set of n planes that contain l (e.g., see Figure 1). This co...

We continue the study of counting complexity begun in [13, 14, 15] by proving upper and lower bounds on the complexity of computing the Hilbert polynomial of a homogeneous ideal. We show that the problem of computing the Hilbert polynomial of a smooth equidimensional complex projective variety can be reduced in polynomial time to the problem of counting the number of complex common zeros of a f...