We prove that any Hecke eigenclass in the mod p cohomology of a congruence subgroup of GL(1,Z) or GL(2,Z) has attached to it a mod p Galois representation such that the characteristic polynomial of a Frobenius element at a prime l equals the Hecke polynomial at l.

Let f (x,y,z) is a cyclic homogeneous polynomial of degree four of three nonnegative real variables satisfying the condition f (1,1,1) = 0 . We find necessary and sufficient condition to be true the inequality f (x,y,z) 0 , for this aim we introduce a characteristic polynomial Jf (t) and by its root t0 > 0 we formulate the condition.

We present several new polynomial identities associated with matroids and geometric lattices, and relate them to formulas for the characteristic polynomial and the Tutte polynomial. The identities imply a formula for the L^ e numbers of complex hyperplane arrangements.

Let W be a real reflection group, and let Lw denote the lattice consisting of all possible intersections of reflecting hyperplanes of reflections in W. Let pw(t) be the characteristic polynomial of Lw. To every element X of Lw there corresponds a parabolic subgroup of W denoted by Gal(X). If W is irreducible, we show that an element X of Lw is modular if and only if pG,~(x)(t) divides pw(t). Th...

We estimate the fraction of isogeny classes of abelian varieties over a finite field which have a given characteristic polynomial P (T ) modulo l. As an application we find the proportion of isogeny classes of abelian varieties with a rational point of order l.

We present a conjectural formula for the principal minors and the characteristic polynomial of Gross’s regulator matrix associated to a totally odd character of a totally real field. The formula is given in terms of the Eisenstein cocycle, which was defined and studied earlier by the authors and collaborators. For the determinant of the regulator matrix, our conjecture follows from recent work ...

Starting from the expression for the superdeterminant of (xI −M), where M is an arbitrary supermatrix , we propose a definition for the corresponding characteristic polynomial and we prove that each supermatrix satisfies its characteristic equation. Depending upon the factorization properties of the basic polynomials whose ratio defines the above mentioned superdeterminant we are able to constr...

We consider the complexities of substitutive sequences over a binary alphabet. By studying various types of special words, we show that, knowing some initial values, its complexity can be completely formulated via a recurrence formula determined by the characteristic polynomial.

Let P be a locally finite poset with the interval space Int(P ), and R a ring with identity. We shall introduce the Möbius conjugation μ∗ sending each function f : P → R to an incidence function μ∗(f) : Int(P ) → R such that μ∗(fg) = μ∗(f) ∗μ∗(g). Taking P to be the intersection poset of a hyperplane arrangement A, we shall obtain a convolution identity for the number r(A) of regions and the nu...

We show that a pair of matrices satisfying a certain algebraic identity, reminiscent of the binomial theorem, must have the same characteristic polynomial. This is a generalization of Problem 4 (11th grade) from the Romanian National Mathematical Olympiad 2011.