3,062 research outputs found
Homomorphism Polynomials Complete for VP
The VP versus VNP question, introduced by Valiant, is probably the most important open question in algebraic complexity theory. Thanks to completeness results, a variant of this question, VBP versus VNP, can be succinctly restated as asking whether the permanent of a generic matrix can be written as a determinant of a matrix of polynomially bounded size. Strikingly, this restatement does not mention any notion of computational model. To get a similar restatement for the original and more fundamental question, and also to better understand the class itself, we need a complete polynomial for VP. Ad hoc constructions yielding complete polynomials were known, but not natural examples in the vein of the determinant. We give here several variants of natural complete polynomials for VP, based on the notion of graph homomorphism polynomials
A Dichotomy Theorem for Homomorphism Polynomials
In the present paper we show a dichotomy theorem for the complexity of
polynomial evaluation. We associate to each graph H a polynomial that encodes
all graphs of a fixed size homomorphic to H. We show that this family is
computable by arithmetic circuits in constant depth if H has a loop or no edge
and that it is hard otherwise (i.e., complete for VNP, the arithmetic class
related to #P). We also demonstrate the hardness over the rational field of cut
eliminator, a polynomial defined by B\"urgisser which is known to be neither VP
nor VNP-complete in the field of two elements, if VP is not equal to VNP (VP is
the class of polynomials computable by arithmetic circuit of polynomial size)
On the R-matrix realization of Yangians and their representations
We study the Yangians Y(a) associated with the simple Lie algebras a of type
B, C or D. The algebra Y(a) can be regarded as a quotient of the extended
Yangian X(a) whose defining relations are written in an R-matrix form. In this
paper we are concerned with the algebraic structure and representations of the
algebra X(a). We prove an analog of the Poincare-Birkhoff-Witt theorem for X(a)
and show that the Yangian Y(a) can be realized as a subalgebra of X(a).
Furthermore, we give an independent proof of the classification theorem for the
finite-dimensional irreducible representations of X(a) which implies the
corresponding theorem of Drinfeld for the Yangians Y(a). We also give explicit
constructions for all fundamental representation of the Yangians.Comment: 65 page
A construction of integer-valued polynomials with prescribed sets of lengths of factorizations
For an arbitrary finite set S of natural numbers greater 1, we construct an
integer-valued polynomial f, whose set of lengths in Int(Z) is S. The set of
lengths of f is the set of all natural numbers n, such that f has a
factorization as a product of n irreducibles in Int(Z)={g in Q[x] | g(Z)
contained in Z}.Comment: To appear in Monatshefte f\"ur Mathematik; 11 page
Galois structure on integral valued polynomials
We characterize finite Galois extensions of the field of rational numbers
in terms of the rings , recently
introduced by Loper and Werner, consisting of those polynomials which have
coefficients in and such that is contained in
. We also address the problem of constructing a basis for as a -module.Comment: final version, accepted for publication in J. Number Theory (2016).
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