In this paper we associate to each multivariate polynomial f that is homogeneous relative to a subset of its variables a series of polynomial families Pλ(f) of m-tuples of homogeneous polynomials of equal degree such that the circuit size of any member in Pλ(f) is bounded from above by the circuit size of f . This provides a method for obtaining lower bounds for the circuit size of f by proving...

In this paper we associate to each multivariate polynomial f that is homogeneous relative to a subset of its variables a series of polynomial families Pλ(f) of m-tuples of homogeneous polynomials of equal degree such that the circuit size of any member in Pλ(f) is bounded from above by the circuit size of f . This provides a method for obtaining lower bounds for the circuit size of f by proving...

The map H → H↓ assigns to each finite-dimensional space of smooth functions a homogeneous polynomial space of the same dimension. We discuss applications of this map in the areas of multivariate polynomial interpolation, box spline theory and polynomial ideals. §

We continue the study of counting complexity begun in [13, 14, 15] by proving upper and lower bounds on the complexity of computing the Hilbert polynomial of a homogeneous ideal. We show that the problem of computing the Hilbert polynomial of a smooth equidimensional complex projective variety can be reduced in polynomial time to the problem of counting the number of complex common zeros of a f...

It is shown that the Jacobian Conjecture holds for all polynomial maps F : k → k of the form F = x + H , such that JH is nilpotent and symmetric, when n ≤ 4. If H is also homogeneous a similar result is proved for all n ≤ 5. Introduction Let F := (F1, . . . , Fn) : C → C be a polynomial map i.e. each Fi is a polynomial in n variables over C. Denote by JF := (i ∂xj )1≤i,j≤n, the Jacobian matrix ...

The multi-homogeneous Bézout number is a bound for the number of solutions of a system of multi-homogeneous polynomial equations, in a suitable product of projective spaces. Given an arbitrary, not necessarily multi-homogeneous system, one can ask for the optimal multi-homogenization that would minimize the Bézout number. In this paper, it is proved that the problem of computing, or even estima...