The main purpose of this work is to propose new notions of equivalence between polynomial matrices, that preserve both the finite and infinite elementary divisor structure. The approach we use is twofold : a) the ”homogeneous polynomial matrix approach” where in place of the polynomial matrices, we study their homogeneous polynomial matrix forms and use 2-D equivalence transformations in order ...

This paper introduces the ulmods, a generalisation of modules for the homogeneous relations. We first present some properties of the ulmod family, then show that, if the homogeneous relation fulfills some natural axioms, the ulmod family has a unique decomposition tree. Under a certain size assumption we show that this tree can be computed in polynomial time. We apply this theory to a new tourn...

In this paper we extract the geometric characteristics from an antipodally symmetric spherical function (ASSF), which can be described equivalently in the spherical harmonic (SH) basis, in the symmetric tensor (ST) basis constrained to the sphere, and in the homogeneous polynomial (HP) basis constrained to the sphere. All three bases span the same vector space and are bijective when the rank of...

This paper introduces the ulmods, a generalisation of modules for the homogeneous relations. We first present some properties of the ulmod family, then show that, if the homogeneous relation fulfills some natural axioms, the ulmod family has a unique decomposition tree. Under a certain size assumption we show that this tree can be computed in polynomial time. We apply this theory to a new tourn...

Let R be a ring and let B be a commutative ring. Let p : R −→ B be a homogeneous multiplicative polynomial law of degree n. The main result of this paper is to show that p is essentially a determinant, in the sense that p is obtained from a determinant by left and right composition with ring homomorphisms. This is achieved using results on the invariants of matrices in positive characteristic d...

We consider polynomial eigenvalue problems P(A, α, β)x = 0 in which the matrix polynomial is homogeneous in the eigenvalue (α, β) ∈ C2. In this framework infinite eigenvalues are on the same footing as finite eigenvalues. We view the problem in projective spaces to avoid normalization of the eigenpairs. We show that a polynomial eigenvalue problem is wellposed when its eigenvalues are simple. W...

Suppose p is a homogeneous polynomial over a field K. Let p(D) be the differential operator defined by replacing each occurrence of xj in p by xj , defined formally in case K is not a subset of C. (The classical example is p(x1 , ..., xn)= j xj , for which p(D) is the Laplacian { .) In this paper we solve the equation p(D) q=0 for homogeneous polynomials q over K, under the restriction that K b...