54687552
Arithmetically exceptional functions and elliptic curves
Let f(X) 2 Q(X) be a rational function. For almost all primes p we can reduce the coecients of f and consider fp := f mod p as a function on the projective line P 1 (Fp) = Fp[ f1g. Here we continue the arithmetic aspects of joint work with Guralnick and Saxl, and classify the functions f such that fp is a bijection for innitely many primes p. This is the rational function analog of the classical conjecture of Schur (1923), solved by Fried (1970), which considered the case that f is a polynomial. Thereby we also answer a question of J. G. Thompson about the minimal eld of denition of a certain rational function of degree 25.
5739920
An infinite series of Kronecker conjugate polynomials
Let K be a field of characteristic 0, t a transcendental over K, and Ir be the absolute Galois group of K(t). Then two non-constant polynomials f, g E K[X] are said to be Kronecker conjugate if an element of F fixes a root of f (X) t if and only if it fixes a root of g(X) t. If K is a number field, and f, g E OK [X] where OK is the ring of integers of K, then f and g are Kronecker conjugate if and only if the value set f((OK) equals g(OK) modulo all but finitely many non-zero prime ideals of O9K. In 1968 H. Davenport suggested the study of this latter arithmetic property. The main progress is due to M. Fried, who showed that under certain assumptions the polynomials f and g differ by a linear substitution. Further, he found non-trivial examples where Kronecker conjugacy holds. Until now there were only finitely many known such examples. This paper provides the first infinite series. The main part of the construction is group theoretic.
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.
123375672
Transitive linear groups with large soluble normal subgroups
17569071
A Weil-Bound Free Proof of Schur's Conjecture
Letfbe a polynomial with coefficients in the ring OKof integers of a number field. Suppose thatfinduces a permutation on the residue fields OK/p for infinitely many nonzero prime ideals p of OK. Then Schur's conjecture, namely thatfis a composition of linear and Dickson polynomials, has been proved by M. Fried. All the present versions of the proof use Weil's bound on the number of points of absolutely irreducible curves over finite fields in order to get a Galois theoretic translation and to finish the proof by means of finite group theory. This note replaces the use of this deep result by elementary arguments.
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
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.