In this paper, we propose a symbolic-numeric algorithm to count the number of solutions zero-dimensional square polynomial system within local region. We show that succeeds under condition region is sufficiently small and well-isolating for k-fold solution z system. our analysis, derive bound on size guarantees success. further argue depends parameters such as norm multiplicity well distances b...

In this paper we give a simple polynomial-time algorithm to exactly count the number of Euler Tours (ETs) of any Eulerian graph of bounded treewidth. The problems of counting ETs are known to be ♯P complete for general graphs (Brightwell and Winkler, 2005 [4]). To date, no polynomial-time algorithm for counting Euler tours of any class of graphs is known except for the very special case of seri...

For every prime m ≥ 2, we give a family of tautologies that require super-polynomial size constant-depth Frege proofs from Countm axioms, and whose algebraic translations have constant-degree, polynomial size polynomial calculus refutations over Zm. This shows that constant-depth Frege systems with counting axioms modulo m do not polynomially simulate constant-depth Frege systems with counting ...

In the analysis of the structure of totally categorical first order theories, the second author showed that certain combinatorial counting functions play an important role. Those functions are invariants of the structures and are always polynomials in one or many variables, depending on the number of independent dimensions of the theory in question. The first author introduced the notion of gra...

We consider certain counting problems involving colourings of graphs and independent sets in hypergraphs. Using polynomial interpolation techniques, we show that these problems are #P -complete. Therefore, e cient approximate counting is the most one can realistically expect to achieve. Rapidly mixing Markov chains which can be used for approximately solving these counting problems have been re...

For a boolean function f : {0, 1} → {0, 1}, let f̂ be the unique multilinear polynomial such that f(x) = f̂(x) holds for every x ∈ {0, 1}. We show that, assuming VP 6= VNP, there exists a polynomialtime computable f such that f̂ requires super-polynomial arithmetic circuits. In fact, this f can be taken as a monotone 2-CNF, or a product of affine functions. This holds over any field. In order to p...

We introduce the notion of a scheduling problem which is a boolean function S over atomic formulas of the form xi ≤ xj . Considering the xi as jobs to be performed, an integer assignment satisfying S schedules the jobs subject to the constraints of the atomic formulas. The scheduling counting function counts the number of solutions to S. We prove that this counting function is a polynomial in t...