121106704
(An,Sn) Realizations by Polynomials–On a Question of Fried
While disproving a conjecture of Cohen about monodromy groups of polynomials and applying this to give new counterexamples to a question of Chowla and Zassenhaus in an earlier paper (1995, M. Fried,Finite Fields Appl.,1, 326?359), Fried asked whether there are polynomials over Q of odd square degreenwith geometric and arithmetic monodromy group the alternating groupAnand symmetric groupSn, respectively. In this note we give two different proofs that such polynomials do not exist.
59486970
Hilbert’s irreducibility theorem for prime degree and general polynomials
Letf (X, t)εℚ[X, t] be an irreducible polynomial. Hilbert’s irreducibility theorem asserts that there are infinitely manyt0εℤ such thatf (X, t0) is still irreducible. We say thatf (X, t) isgeneral if the Galois group off (X, t) over ℚ(t) is the symmetric group in its natural action. We show that if the degree off with respect toX is a prime ≠ 5 or iff is general of degree ≠ 5, thenf (X, t0) is irreducible for all but finitely manyt0εℤ unless the curve given byf (X, t)=0 has infinitely many points (x0,t0) withx0εℚ,t0εℤ. The proof makes use of Siegel’s theorem about integral points on algebraic curves, and classical results about finite groups, going back to Burnside, Schur, Wielandt, and others.
122256711
Exceptional Polynomials of Affine Type
Abstract Let K be a finite field of characteristic p . A polynomial f with coefficients in K is said to be exceptional if it induces a permutation on infinitely many finite extensions of K . Let t be a transcendental, and K be an algebraic closure of K . The exceptional polynomials known to date are constructed from certain twists of polynomials g , such that the splitting field of g ( X ) −  t over K is rational. On the other hand we know that except for p  = 2 or 3, either the degree of an exceptional polynomial f is a power of the characteristic p , or f is a Dickson polynomial. The case deg  f  =  p has been settled completely. We give a new series of exceptional polynomials of degree p m with m even which does not follow the above mentioned construction principle. We show under certain additional hypotheses that the associated monodromy groups of exceptional polynomials will be severely restricted. In particular, we determine precisely what the geometric monodromy groups of exceptional polynomials of degree p 2 will be.
15736393
Kronecker conjugacy of polynomials
Let f, g ∈ Z[X] be non-constant polynomials with integral coefficients. In 1968 H. Davenport raised the question as to when the value sets f(Z) and g(Z) are the same modulo all but finitely many primes. The main progress until now is M. Fried’s result that f and g then differ by a linear substitution, provided that f is functionally indecomposable. We extend this result to polynomials f of composition length 2. Also, we study the analog when Z is replaced by the integers of a number field. The above number theoretic property translates to an equivalent property of subgroups of a finite group, known as Kronecker conjugacy, a weakening of conjugacy which has been studied by various authors under different assumptions and in other contexts. We also give a simplified and strengthened version of the Galois theoretic translation to finite groups.
123589108
On simple semiabelian p-adic lie algebras
121536074
Sharply transitive linear algebraic groups
We study Zariski-closed linear groupsG ≤ GLn(k) over fieldsk of characteristic 0 which act sharply transitively on the non-zero vectors ofkn. For square-freen, orn⩽15, or ifk has cohomological dimension ⩽1 we obtain a complete classification (i.e. a reduction to questions about associative division algebras). The main tools are representation theory of Lie algebras over algebraically closed and non-closed fields, and results about simple associative algebras in order to control the interplay between linear Lie algebras and the associative algebras generated by them. The relation to nearfields and left-symmetric division algebras is also discussed.
122312791
STOCHASTIC FORCING OF OCEANIC MOTIONS
Most oceanic motions are forced by stresses and heat and fresh water fluxes at the air-sea interface. When these forcing fields are assumed to be prescribed and stochastic and when the oceanic response is assumed to be linear then the forcing problem reduces to a set of stochastically forced linear oscillators. This set can in principle be decoupled. The asymptotic response of a linear oscillator to stationary random forcing is well understood and depends on whether the oscillator is stable, unstable or neutral. The explicit decoupling requires additional simplifying assumptions as exemplified by the stochastic forcing of surface gravity waves, internal gravity waves, Rossby waves and sea-surface temperature anomalies A powerful diagnostic tool is a coherence map which desribes the coherence between the oceanic response at one location and the atmospheric forcing at another location as a function of separation for different frequancies. A simple model of the stochastic forcing of barotropic Rossby waves by fluctuations in the atmospheric windstress reproduces basic features of observed coherence maps. The model expecially accounts for the qualitative changes that occur when different oceanic variables are considered or when the frequency is changed.
117137965
Stochastic forcing of quasi-geostrophic eddies
The basic theoretical framework is presented for the study of the stochastic atmospheric forcing of barotropic quasi-geostrophic oceanic eddies. The governing equation is the linearized potential vorticity equation with a stationary stochastic forcing function representing random fluctuations in the atmospheric windstress curl. The frequency autospectrum of the response and the frequency cross-spectrum between forcing and response are spatial convolutions of the frequency spectrum of the forcing with the Green’s function of the potential vorticity equation. Coherence maps have been calculated for simple geometries (meridional channel or infinite ocean) and simple forcing spectra (Gaussian or white in wavenumber space). They reproduce basic features of observed coherence maps. The major open question is whether the more complex structure of the observed maps is due to the more complex structure of the forcing spectrum or due to the processes neglected in the potential vorticity equation.
118540313
On the diffusion of momentum and mass by internal gravity waves