The theory of constant polynomial combinants has been well developed [2] and it is linked to the linear part of the constant Determinantal Assignment problem [1] that provides the unifying description of the pole and zero assignment problems in Linear Systems. Considering the case of dynamic pole, zero assignment problems leads to the emergence of dynamic polynomial combinants. This paper aims ...

It is very well-known that this system has a non-trivial solution if and only if AD −BC equals to zero. One can generalize the previous situation in two different directions. The most classical one is the notion of determinant, which is the condition under which a system of n homogeneous equations in n unknowns    A11x1 + A12x2 + . . . + A1nxn = 0 A21x1 + A22x2 + . . . + A2nxn = 0 .. .. ...

Our first contribution is a substantial acceleration of randomized computation of scalar, univariate, and multivariate matrix determinants, in terms of the output-sensitive bit operation complexity bounds, including computation modulo a product of random primes from a fixed range. This acceleration is dramatic in a critical application, namely solving polynomial systems and related studies, via...

We consider the interplay of point counts, singular cohomology, étale cohomology, eigenvalues of the Frobenius and the Grothendieck ring of varieties for two families of varieties: spaces of rational maps and moduli spaces of marked, degree d rational curves in P. We deduce as special cases algebro-geometric and arithmetic refinements of topological computations of Segal, Cohen–Cohen–Mann–Milgr...

Let W be a vector space of dimension n+ 1 over a field K. The Chow divisor of a k-dimensional variety X in P = P(W ) is the hypersurface, in the Grassmannian Gk+1 of planes of codimension k+1 in P, whose points (over the algebraic closure of K) are the planes that meet X . The Chow form of X is the defining equation of the Chow divisor. For example, the resultant of k+ 1 forms of degree e in k ...

This paper constructs the multipole expansion (in general relativity) of the gravitational field generated by a slowly-moving isolated source. We introduce some definitions for the source multipole moments, valid to all orders in a post-Newtonian expansion, and depending in a well-defined way on the total stress-energy pseudo-tensor of the material and gravitational fields. Previously obtained ...