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.
118540313
On the diffusion of momentum and mass by internal gravity waves