Composite Polynomials with Coefficients in an Arbitrary Field of Characteristic Zero

1942 ◽  
Vol 64 (1/4) ◽  
pp. 389 ◽  
Author(s):  
Howard Levi
Keyword(s):  
Characteristic Zero ◽  
Arbitrary Field

XV.—Bounds for the Indices of Nilpotent and Solvable Lie Algebras

1954 ◽  
Vol 64 (2) ◽  
pp. 200-208
Author(s):  
E. M. Patterson
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Characteristic Zero ◽  
Arbitrary Field ◽  
Nilpotent Lie Algebras ◽  
Solvable Lie Algebras

SynopsisBy examining certain connections between the derivatives and the powers of a Lie algebra, bounds are obtained for the indices of nilpotent Lie algebras over an arbitrary field. The results are used to obtain bounds for the indices of solvable Lie algebras over a field of characteristic zero.

Rationality of the quotient of ℙ2 by finite group of automorphisms over arbitrary field of characteristic zero

2014 ◽  
Vol 12 (2) ◽  
Author(s):  
Andrey Trepalin
Keyword(s):  
Finite Group ◽  
Projective Plane ◽  
Characteristic Zero ◽  
Arbitrary Field ◽  
Group Of Automorphisms ◽  
A Finite Group

AbstractLet $$\Bbbk$$ be a field of characteristic zero and G be a finite group of automorphisms of projective plane over $$\Bbbk$$. Castelnuovo’s criterion implies that the quotient of projective plane by G is rational if the field $$\Bbbk$$ is algebraically closed. In this paper we prove that $${{\mathbb{P}_\Bbbk ^2 } \mathord{\left/ {\vphantom {{\mathbb{P}_\Bbbk ^2 } G}} \right. \kern-\nulldelimiterspace} G}$$ is rational for an arbitrary field $$\Bbbk$$ of characteristic zero.

scholarly journals Submodules of Specht modules for Weyl groups

1996 ◽  
Vol 39 (1) ◽  
pp. 43-50
Author(s):  
Saіt Halicioğlu
Keyword(s):  
Characteristic Zero ◽  
Symmetric Groups ◽  
Root Systems ◽  
Weyl Groups ◽  
Arbitrary Field ◽  
Specht Modules ◽  
Irreducible Modules

The construction of all irreducible modules of the symmetric groups over an arbitrary field which reduce to Specht modules in the case of fields of characteristic zero is given by G. D. James. Halicioğlu and Morris describe a possible extension of James' work for Weyl groups in general, where Young tableux are interpreted in terms of root systems. In this paper we show how to construct submodules of Specht modules for Weyl groups.

Teissier's problem on inequalities of nef divisors

2015 ◽  
Vol 14 (09) ◽  
pp. 1540002 ◽  
Author(s):  
Steven Dale Cutkosky
Keyword(s):  
Characteristic Zero ◽  
Intersection Theory ◽  
Line Bundles ◽  
Closed Field ◽  
Arbitrary Field ◽  
Riemann Manifold ◽  
Complete Proof ◽  
Complete Variety ◽  
Direct System ◽  
Nef Divisors

Teissier has proven remarkable inequalities [Formula: see text] for intersection numbers si = (ℒi ⋅ ℳd-i) of a pair of nef line bundles ℒ, ℳ on a d-dimensional complete algebraic variety over a field. He asks if two nef and big line bundles are numerically proportional if the inequalities are all equalities. In this paper, we show that this is true in the most general possible situation, for nef and big line bundles on a proper irreducible scheme over an arbitrary field k. Boucksom, Favre and Jonsson have recently established this result on a complete variety X over an algebraically closed field of characteristic zero. Their proof involves an ingenious extension of the intersection theory on a variety to its Zariski Riemann Manifold. This extension requires the existence of a direct system of nonsingular varieties dominating X. We make use of a simpler intersection theory which does not require resolution of singularities, and extend volume to an arbitrary field and prove its continuous differentiability, extending results of Boucksom, Favre and Jonsson, and of Lazarsfeld and Mustaţă. A goal in this paper is to provide a manuscript which will be accessible to many readers. As such, subtle topological arguments which are required to give a complete proof in [S. Bouksom et al., J. Algebraic Geometry18 (2009) 279–308] have been written out in this manuscript, in the context of our intersection theory, and over arbitrary varieties.

scholarly journals Double Points of Free Projective Line Arrangements

2020 ◽  
Author(s):  
Takuro Abe
Keyword(s):  
General Result ◽  
Characteristic Zero ◽  
Double Point ◽  
Projective Line ◽  
Arbitrary Field ◽  
Open Problems ◽  
Line Arrangement ◽  
Double Points ◽  
Line Arrangements ◽  
Arrangements Of Lines

Abstract We prove the Anzis–Tohăneanu conjecture, that is, the Dirac–Motzkin conjecture for supersolvable line arrangements in the projective plane over an arbitrary field of characteristic zero. Moreover, we show that a divisionally free arrangements of lines contain at least one double point that can be regarded as the Sylvester–Gallai theorem for some free arrangements. This is a corollary of a general result that if you add a line to a free projective line arrangement, then that line has to contain at least one double point. Also, we prove some conjectures and one open problems related to supersolvable line arrangements and the number of double points.

Short Generating Functions for some Semigroup Algebras

10.37236/1729 ◽  
2003 ◽  
Vol 10 (1) ◽  
Author(s):  
Graham Denham
Keyword(s):  
Rational Function ◽  
Generating Functions ◽  
Hilbert Series ◽  
Simple Expression ◽  
Arbitrary Field ◽  
Graded Algebra ◽  
Semigroup Algebras ◽  
Positive Integers

Let $a_1,\ldots,a_n$ be distinct, positive integers with $(a_1,\ldots,a_n)=1$, and let k be an arbitrary field. Let $H(a_1,\ldots,a_n;z)$ denote the Hilbert series of the graded algebra k$[t^{a_1},t^{a_2},\ldots,t^{a_n}]$. We show that, when $n=3$, this rational function has a simple expression in terms of $a_1,a_2,a_3$; in particular, the numerator has at most six terms. By way of contrast, it is known that no such expression exists for any $n\geq4$.

scholarly journals On the Voevodsky motive of the moduli space of Higgs bundles on a curve

2021 ◽  
Vol 27 (1) ◽  
Author(s):  
Victoria Hoskins ◽  
Simon Pepin Lehalleur
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
Characteristic Zero ◽  
Rational Point ◽  
Triangulated Category ◽  
Higgs Bundles ◽  
Projective Curve ◽  
Smooth Projective Curve ◽  
De Rham

AbstractWe study the motive of the moduli space of semistable Higgs bundles of coprime rank and degree on a smooth projective curve C over a field k under the assumption that C has a rational point. We show this motive is contained in the thick tensor subcategory of Voevodsky’s triangulated category of motives with rational coefficients generated by the motive of C. Moreover, over a field of characteristic zero, we prove a motivic non-abelian Hodge correspondence: the integral motives of the Higgs and de Rham moduli spaces are isomorphic.

scholarly journals Irreducible modules with highest weight vectors over modular Witt and special Lie superalgebras

2019 ◽  
Vol 17 (1) ◽  
pp. 1381-1391
Author(s):  
Keli Zheng ◽  
Yongzheng Zhang
Keyword(s):  
Lie Superalgebras ◽  
Arbitrary Field ◽  
Special Linear ◽  
Highest Weight ◽  
Irreducible Modules

Abstract Let 𝔽 be an arbitrary field of characteristic p > 2. In this paper we study irreducible modules with highest weight vectors over Witt and special Lie superalgebras of 𝔽. The same irreducible modules of general and special linear Lie superalgebras, which are the 0-th part of Witt and special Lie superalgebras in certain ℤ-grading, are also considered. Then we establish a certain connection called a P-expansion between these modules.

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