We give a uniform method for the two problems of counting the connected and irreducible components of complex algebraic varieties. Our algorithms are purely algebraic, i.e., they use only the field structure of C. They work in parallel polynomial time, i.e., they can be implemented by algebraic circuits of polynomial depth. The design of our algorithms relies on the concept of algebraic differe...

In the search for a logic capturing polynomial time most promising candidates are Choiceless Polynomial Time (CPT) and rank logic. Rank extends fixed-point with counting by operator over prime fields. We show that isomorphism problem CFI graphs ℤ 2 i cannot be defined in logic, even if base graph is totally ordered. However, CPT can define this problem. thereby separate from particular time.

Harary et al. put forward the concept on minimum cardinality over all subsets of perfect matching M that are not included by any other ones, to be forcing number for M. A counting polynomial matchings possessing same was introduced Zhang al., using name `forcing polynomial'. This research deduces recurrence formulas polynomials monotonic constructable hexagonal systems and with one turning. Fro...

Let A be any subspace arrangement in R defined over the integers and let Fq denote the finite field with q elements. Let q be a large prime. We prove that the characteristic polynomial /(A, q) of A counts the number of points in Fq that do not lie in any of the subspaces of A, viewed as subsets of Fq . This observation, which generalizes a theorem of Blass and Sagan about subarrangements of the...

The number of triangulations of a planar n point set is known to be c, where the base c lies between 2.43 and 30. The fastest known algorithm for counting triangulations of a planar n point set runs in O∗(2n) time. The fastest known arbitrarily close approximation algorithm for the base of the number of triangulations of a planar n point set runs in time subexponential in n. We present the firs...

Ad Ilenia, la persona che più di tutte ha cambiato la mia vita. INTRODUCTION Combinatorial counting problems have a long and distinguished history. Apart from their intrinsic interest, they arise naturally from investigations in numerous branches of mathematics and natural sciences and have given rise to a rich and beautiful theory. Ranking problems, which consist in determining the position of...

The number of triangulations of a planar n point set S is known to be c, where the base c lies between 8.65 and 30. Similarly, the number of spanning trees on S is known to be d, where the base d lies between 12.52 and 141.07. The fastest known algorithm for counting triangulations of S runs in O∗(2n) time while that for counting spanning trees on S enumerates them in Ω(12.52) time. The fastest...

Numerous problems in program analysis, can be reduced to finding bounds on the number of integer points in a convex set, or the solution of a more general polyhedral counting problem. For a large class of applications the solution of such a counting problem can be expressed as a piecewise Ehrhart quasi-polynomial in the parameters. This work presentsmethods to find bounds on quasi-polynomials o...

Toda proved a remarkable connection between the polynomial hierarchy and the counting classes. Tarui improved Toda’s result to show the connection to a weak form of counting and provided an elegant proof. This paper shows that a key step in Tarui’s proof can be done uniformly using the depth-first traversal and provides the algorithm that generalizes Toda’s result to arbitrary alternating Turin...

We show that the class of chordal claw-free graphs admits LREC=-definable canonization. LREC= is a logic that extends first-order logic with counting by an operator that allows it to formalize a limited form of recursion. This operator can be evaluated in logarithmic space. It follows that there exists a logarithmic-space canonization algorithm, and therefore a logarithmic-space isomorphism tes...