Let T N; p;k (x) be the characteristic polynomial of the Hecke operator Tp acting on the space of cusp forms Sk(N; ). We describe the factorization of T N; p;k (x) mod ` as k varies, and we explicitly calculate those factorizations for N = 1 and small `. These factorizations are used to deduce the irreducibility of certain T 1;1 q;k (x) from the irreducibility of T 1;1 2;k (x).

The study of ideals in polynomial rings over fields is absolutely central to algebraic geometry. We think of an affine variety (usually called affine algebraic set without an irreducibility hypothesis) in n dimensions as defined to be the common zero set of a collection of polynomials f1(x1, . . . , xn), . . . , fm(x1, . . . , xn) with coefficients in some field k. We see that if a point (c1, ....

We study irreducibility of families of degree 4 Del Pezzo surface fibrations over curves.

In this paper, we study irreducibility in RNA structures. By RNA structure, we mean RNA secondary as well as RNA pseudoknot structures as abstract contact structures. We give an analysis contrasting random and minimum free energy (mfe) configurations and secondary versus pseudoknots structures. In the process, we compute various distributions: the numbers of irreducible substructures and their ...

Given a polynomial P in several variables over an algebraically closed field, we show that except in some special cases that we fully describe, if one coefficient is allowed to vary, then the polynomial is irreducible for all but at most deg(P ) − 1 values of the coefficient. We more generally handle the situation where several specified coefficients vary.