Given a random n-variate degree-d homogeneous polynomial f , we study the following problem:

Solving polynomial systems arising from applications is frequently made easier by the structure of the systems. Weighted homogeneity (or quasi-homogeneity) is one example of such a structure: given a system of weights W = (w1, . . . , wn), W -homogeneous polynomials are polynomials which are homogeneous w.r.t the weighted degree degW (X α1 1 . . . X αn n ) = ∑ wiαi. Gröbner bases for weighted h...

The purpose of this paper is to introduce the norm decomposition which enables us to compute the roots of a monic irreducible imprimitive polynomial f ∈ Q[t] by solving polynomial equations of lower degree. We call an irreducible polynomial f imprimitive if the number field generated by a root of f contains non-trivial subfields. We will see that for each subfield there exists a norm decomposit...

The expected number of real projective roots orthogonally invariant random homogeneous polynomial systems is known to be equal the square root Bézout number. A similar result for multi-homogeneous systems, through a product orthogonal groups. In this note, those results are generalized certain families sparse with no invariance assumed.

The Bertrand-Darboux integrability condition for a certain class of perturbed harmonic oscillators is studied from the viewpoint of the BirkhoffGustavson(BG)-normalization: In solving an inverse problem of the BGnormalization on computer algebra, it is shown that if the perturbed harmonic oscillators with a homogeneous cubic-polynomial potential and with a homogeneous quartic-polynomial potenti...

We construct algebraically closed fields containing an algebraic closure of the field of power series in several variables over a characteristic zero field. Each of these fields depends on the choice of an Abhyankar valuation and are constructed via the Newton-Puiseux method. Then we study more carefully the case of monomial valuations and we give a result generalizing the Abhyankar-Jung Theore...

We complete the complexity classification by degree of minimizing a polynomial in two variables over the integer points in a polyhedron. Previous work shows that in two variables, optimizing a quadratic polynomial over the integer points in a polyhedral region can be done in polynomial time, while optimizing a quartic polynomial in the same type of region is NP-hard. We close the gap by showing...

In this paper, we establish hardness and approximation results for various Lp–ball constrained homogeneous polynomial optimization problems, where p ∈ [2,∞]. Specifically, we prove that for any given d ≥ 3 and p ∈ [2,∞], both the problem of optimizing a degree–d homogeneous polynomial over the Lp–ball and the problem of optimizing a degree–d multilinear form (regardless of its super–symmetry) o...

The aim of this short note is to draw attention to a method by which the partition function and marginal probabilities for a certain class of random fields on complete graphs can be computed in polynomial time. This class includes Ising models with homogeneous pairwise potentials but arbitrary (inhomogeneous) unary potentials. Similarly, the partition function and marginal probabilities can be ...

We propose a polynomial algorithm for linear programming. The algorithm represents a linear optimization or decision problem in the form of a system of linear equations and non-negativity constraints. Then it uses a procedure that either finds a solution for the respective homogeneous system or provides the information based on which the algorithm rescales the homogeneous system so that its fea...