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Vol.28 (2) 2006 — 2 results found.

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The Noether Number in Invariant Theory
C. R. Math. Rep. Acad. Sci. Canada Vol. 28 (2) 2006, pp. 39–62
David L. Wehlau (Received: 2006/02/06, Revised: 2006/06/08)

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Let \(\mathbb{F}\) be any field. Let \(G\) be any reductive linear algebraic group and consider a finite dimensional rational representation \(V\) of \(G\). Then the \(\mathbb{F}\)-algebra \(\mathbb{F}[V]^G\) of polynomial invariants for \(G\) acting on \(V\) is finitely generated. The Noether Number \(\beta(G,V)\) is the highest degree of an element of a minimal homogeneous generating set for \(\mathbb{F}[V]^G\). We survey what is known about Noether Numbers, in particular describing various upper and lower bounds for them. Both finite and infinite groups and both characteristic 0 and positive characteristic are considered.

Soit \(\mathbb{F}\) un corps commutatif. Soit \(G\) un groupe algébrique linéaire réductif, et \(V\) une représentation rationelle de dimension finie sur \(\mathbb{F}\). Alors \(\mathbb{F}[V]^G\), l’anneau des polynômes invariants pour l’action de \(G\) sur \(V\), admet un nombre fini de générateurs. Le nombre de Noether \(\beta(G,V)\) est le degré maximal d’un membre d’un ensemble minimal de générateurs homogènes de \(\mathbb{F}[V]^G\). Nous faisons une revue des résultats connus sur les nombres de Noether. En particulier, nous décrivons certaines bornes supérieures et inférieures pour les nombres de Noether. Nous considérons à la fois les groupes finis et infinis, sur des corps de charactéristique \(0\) ou \(p>0\).

On $L^{(r+1)}(\pi,1/2)$
C. R. Math. Rep. Acad. Sci. Canada Vol. 28 (2) 2006, pp. 33–38
Amir Akbary (Received: 2005/11/15)

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Let \(r\) be the order of vanishing of the automorphic \(L\)-function \(L(\pi,s)\) at \(s=1/2\). We study the non-vanishing of the derivative of order \(r+1\) of \(L(\pi,s)\) at \(s=1/2\).

Soit \(r\) l’ordre d’annulation de la fonction \(L\) automorphe \(L(\pi,s)\) à \(s=1/2\). Nous étudions la non-annulation de la dérivée d’ordre \(r+1\) de \(L(\pi,s)\) à \(s=1/2\).

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algebraic number theory approximation property automorphisms Bessel functions Boson-fermion correspondence C*-algebra Carmichael number center problem Chebyshev transform Classification of simple C*-algebras composition operators continued fractions Cuntz Semigroup cycles of ideals elliptic curves fixed point Fourier transform function fields. general relativity generic property ideals indefinite inner product inductive limits of sub-homogeneous C*- algebras Irrational rotation algebra J-Hermitian matrix K-theory Kahler manifolds L-functions maximal ideal space nonexpansive mapping noninterlacing numerical range orthogonal polynomials Predual space prime number property SP quadratic forms Renormalization rotation algebras Salem number semi-reciprocal polynomials tracially approximate splitting interval algebras unbounded traces Weak Markov set Whitney problems

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05C05 11A07 11A55 11B37 11B68 11D09 11D25 11D41 11E04 11F67 11G05 11R09 11R11 13B25 14J26 14M25 14P10 17B37 17B67 19K56 26A51 30C15 30H05 35B 37E10 37E20 37F20 37F25 39B72 42C05 43A07 43A62 46B20 46L05 46L35 46L40 46L55 46L80 47H10 53B25 53C55 54C60 60F10 60J75 83C05

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