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Elena Angelini Francesco Galuppi Massimiliano Mella Giorgio Ottaviani 《Journal of Pure and Applied Algebra》2018,222(4):950-965
We prove that a general polynomial vector in three homogeneous variables of degrees has a unique Waring decomposition of rank 7. This is the first new case we are aware of, and likely the last one, after five examples known since the 19th century and the binary case. We prove that there are no identifiable cases among pairs in three homogeneous variables of degree , unless , and we give a lower bound on the number of decompositions. The new example was discovered with Numerical Algebraic Geometry, while its proof needs Nonabelian Apolarity. 相似文献
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Let e be a positive integer, p be an odd prime, , and be the finite field of q elements. Let . The graph is a bipartite graph with vertex partitions and , and edges defined as follows: a vertex is adjacent to a vertex if and only if and . If and , the graph contains no cycles of length less than eight and is edge-transitive. Motivated by certain questions in extremal graph theory and finite geometry, people search for examples of graphs containing no cycles of length less than eight and not isomorphic to the graph , even without requiring them to be edge-transitive. So far, no such graphs have been found. It was conjectured that if both f and g are monomials, then no such graphs exist. In this paper we prove the conjecture. 相似文献
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《International Journal of Approximate Reasoning》2014,55(9):1830-1842
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Let be a field of q elements, where q is a power of an odd prime p. The polynomial defined by has the property that where ρ is the quadratic character on . This univariate identity was applied to prove a recent theorem of N. Katz. We formulate and prove a bivariate extension, and give an application to quadratic residuacity. 相似文献
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A Steiner 2- trade is a pair of disjoint partial Steiner triple systems, each on the same set of points, such that each pair of points occurs in if and only if it occurs in . A Steiner 2- trade is called d-homogeneous if each point occurs in exactly d blocks of (or ). In this paper we construct minimal d-homogeneous Steiner 2- trades of foundation and volume for sufficiently large values of . (Specifically, if is divisible by 3 and otherwise.) 相似文献
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Neil J.Y. Fan Peter L. Guo Grace L.D. Zhang 《Journal of Pure and Applied Algebra》2017,221(1):237-250
Parabolic R-polynomials were introduced by Deodhar as parabolic analogues of ordinary R-polynomials defined by Kazhdan and Lusztig. In this paper, we are concerned with the computation of parabolic R-polynomials for the symmetric group. Let be the symmetric group on , and let be the generating set of , where for , is the adjacent transposition. For a subset , let be the parabolic subgroup generated by J, and let be the set of minimal coset representatives for . For in the Bruhat order and , let denote the parabolic R-polynomial indexed by u and v. Brenti found a formula for when , and obtained an expression for when . In this paper, we provide a formula for , where and i appears after in v. It should be noted that the condition that i appears after in v is equivalent to that v is a permutation in . We also pose a conjecture for , where with and v is a permutation in . 相似文献
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Let be an algebraically closed field of characteristic 0, and a Cohen–Macaulay graded domain with . If A is semi-standard graded (i.e., A is finitely generated as a -module), it has the h-vector, which encodes the Hilbert function of A. From now on, assume that . It is known that if A is standard graded (i.e., ), then A is level. We will show that, in the semi-standard case, if A is not level, then divides . Conversely, for any positive integers h and n, there is a non-level A with the h-vector . Moreover, such examples can be constructed as Ehrhart rings (equivalently, normal toric rings). 相似文献
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Haïm Brezis Petru Mironescu 《Annales de l'Institut Henri Poincaré (C) Analyse Non Linéaire》2018,35(5):1355-1376
We investigate the validity of the Gagliardo–Nirenberg type inequality
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with . Here, are non negative numbers (not necessarily integers), , and we assume the standard relationsBy the seminal contributions of E. Gagliardo and L. Nirenberg, (1) holds when are integers. It turns out that (1) holds for “most” of values of , but not for all of them. We present an explicit condition on which allows to decide whether (1) holds or fails. 相似文献
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In this paper we define odd dimensional unitary groups . These groups contain as special cases the odd dimensional general linear groups where R is any ring, the odd dimensional orthogonal and symplectic groups and where R is any commutative ring and further the first author's even dimensional unitary groups where is any form ring. We classify the E-normal subgroups of the groups (i.e. the subgroups which are normalized by the elementary subgroup ), under the condition that R is either a semilocal or quasifinite ring with involution and . Further we investigate the action of by conjugation on the set of all E-normal subgroups. 相似文献
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A.S. Sivatski 《Journal of Pure and Applied Algebra》2018,222(3):560-567
Let F be a field of characteristic distinct from 2, a quadratic field extension. Let further f and g be quadratic forms over L considered as polynomials in n variables, , their matrices. We say that the pair is a k-pair if there exist such that all the entries of the upper-left corner of the matrices and are in F. We give certain criteria to determine whether a given pair is a k-pair. We consider the transfer determined by the -linear map with , , and prove that if , then is a -pair. If, additionally, the form does not have a totally isotropic subspace of dimension over , we show that is a -pair. In particular, if the form is anisotropic, and , then is a k-pair. 相似文献
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Ramachandran Balasubramanian Bruno Calado Hervé Queffélec 《Comptes Rendus Mathematique》2006,342(1):7-10
We extend to the setting of Dirichlet series previous results of Bohr for Taylor series in one variable, themselves generalized by Paulsen, Popescu and Singh or extended to several variables by Aizenberg, Boas and Khavinson. We show in particular that, if , with , then and even slightly better, and , C being an absolute constant. To cite this article: R. Balasubramanian et al., C. R. Acad. Sci. Paris, Ser. I 342 (2006). 相似文献
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