M. Pisonero

Invariant factors of an endomorphism and finite free resolutions

Polynomial invariants are associated to an endomorphism u of a module M that has a finite free resolution. The first invariant χ(u, X) (characteristic polynomial of u) is a monic polynomial of degree the Euler characteristic of M. Its construction is based on the MacRae invariant. When M is a finite free module, χ(u, X) is the classical characteristic polynomial of u. With additional assumptions there is constructed a finite sequence of monic polynomials {di(u, X)}i ⩾ 0 such that their product is χ(u, X) and di(u, X) divides di + 1(u, X). When R is a field, these polynomials are the invariant factors of u. A generalized Cayley-Hamilton theorem is given. The generic behavior of the polynomials χ(u, X) and {di(u, X)}i ⩾ 0 in Spec R is proved. Finally it is shown, under certain assumptions that there exists a free submodule F of M, invariant with respect to u, such that the restriction of u to F is similar to the endomorphism of F defined by the diagonal block matrix where the ith block is the companion matrix of di(u, X).

Monomial Inequalities for Newton Coefficients and Determinantal Inequalities for p-Newton Matrices

We consider Newton matrices for which the Newton coefficients are positive. We show that one monomial in these coefficients dominates another for all such Newton matrices if and only if a certain generalized form of majorization occurs. As the Newton coefficients may be viewed as average values of principal minors of a given size, these monomial inequalities may be interpreted as determinantal inequalities in such familiar classes as the positive definite, totally positive, and M-matrices, etc.

Symmetric nonnegative realizability via partitioned majorization

A sufficient condition for symmetric nonnegative realizability of a spectrum is given in terms of (weak) majorization of a partition of the negative eigenvalues by a selection of the positive eigenvalues. If there are more than two positive eigenvalues, an additional condition, besides majorization, is needed on the partition. This generalizes observations of Suleǐmanova and Loewy about the cases of one and two positive eigenvalues, respectively. It may be used to provide insight into realizability of 5-element spectra and beyond.

What polynomial satisfies a given endomorphism

Abstract The Authors study the ideal I(u) formed by the polynomial relations satisfied by an endomorphism u of a module M that has a finite free resolution. When M is torsion free, the ideal I(u) is characterized in terms of the characteristic polynomial χ(u,X) of u. This result gives a natural generalization of McCoys theorem. The ideal I(u) is also related to the invariant factors of u. It is proved that in a large class of rings the ideal I(u) is principally generated by the last proper invariant factor of u. One obtains, for these cases, complete similarity with the classical case of fields.

Determinants on modules that have a finite free resolution

This paper is devoted to define and study the determinant of an endomorphism of a module that has a finite free resolution. It is shown that many of the properties are the same as in free case A relation is given between the character (surjective or injective) of an endomorphism and the character (unit or non zerodivision) of its determinant. It is proved that when the module is torsion free the parallelism with the free case is complete.

On spectra of weighted graphs of order ≤5

Abstract The problem of characterizing the real spectra of weighted graphs is only solved for weighted graphs of order n ≤ 4 . We overview these known results, that come from the context of nonnegative matrices, and give a new method to rule out many unresolved spectra of size 5.

Comparison on the spectral radii of weighted digraphs that differ in a certain subdigraph

Abstract Let D S be a weighted digraph of order n with a subdigraph S of order k, M ( D S ) its adjacency weight matrix and ρ ( D S ) its spectral radius. We consider the class C k of weighted digraphs of order k and we study the preorder in C k given by D S ′ ≾ D S if and only if ρ ( D S ′ ) ≤ ρ ( D S ) . We obtain that this order is equivalent to the entry-wise order M ( D S ′ ) ≤ M ( D S ) . Several points of view are taken, under varying regularity conditions, and k polynomial conditions for the comparison are presented.

On Sufficient Conditions for the RNIEP and their Margins of Realizability

The Real Nonnegative Inverse Eigenvalue Problem (RNIEP) is that of characterizing all possible real spectra of nonnegative matrices. In this work we list some inclusion relations between several sufficient conditions and we study the negativity and the realizability margin of a spectrum with respect to these conditions.