We show that a problem of deciding whether a formula for a multivariate polynomial of n variables over a finite field of characteristic 2 has degree n when reduced modulo a certain Boolean ideal belongs to ⊕ P. When the formula is allowed to have succinct representations as sums of monomials, the problem becomes ⊕ P-complete. key words: degree of multivariate polynomials, finite field of charac...

Given a valuation on the function field k(x, y), we examine the set of images of nonzero elements of the underlying polynomial ring k[x, y] under this valuation. For an arbitrary field k, a Noetherian power series is a map z : Q → k that has Noetherian (i.e., reverse well-ordered) support. Each Noetherian power series induces a natural valuation on k(x, y). Although the value groups correspondi...

While shorter characteristic path length has in general been believed to enhance synchronizability of a coupled oscillator system on a complex network, the suppressing tendency of the heterogeneity of the degree distribution, even for shorter characteristic path length, has also been reported. To see this, we investigate the effects of various factors such as the degree, characteristic path len...

Let Fq be a finite field with q elements, where q is the power of an odd prime, and let GSp(2n, Fq) and GO±(2n, Fq) denote the symplectic and orthogonal groups of similitudes over Fq, respectively. We prove that every real-valued irreducible character of GSp(2n, Fq) or GO±(2n, Fq) is the character of a real representation, and we find the sum of the dimensions of the real representations of eac...

Decision trees constructed by ID3-like algorithms suffer from an inability of detecting instances of categories not present in the set of training examples, i.e., they are discriminative representations. Instead, such instances are assigned to one of the classes actually present in the training set, resulting in undesired misclassifications. In this report, two methods of reducing this problem ...

The formula mentioned in the title is proved. Introduction Let S, T be complete nonsingular surfaces over an algebraically closed field k of any characteristic, and let h : T → S be a finite separable morphism of degree n. We establish a formula that expresses the Euler characteristic (understood as the degree of the second Chern class ∫ c2,T ) of T via the Euler characteristic of S and some lo...

Article history: Received 12 February 2010 Accepted 18 July 2010 Available online xxxx Communicated by Rudolf Lidl

Let Mn(q) be the set of all n× n matrices with entries in the finite field Fq. With asymptotic probability one, the characteristic polynomial of a random A ∈ Mn(q) does not have all its roots in Fq. Let Xn(A) be the degree of the splitting field of the characteristic polynomial of A, and let μn be the average degree: μn = 1 |Mn(q)| ∑