On Sign Conditions Over Real Multivariate Polynomials

On sign conditions over real multivariate polynomials

We present a new probabilistic algorithm to find a finite set of points intersecting the closure of each connected component of the realization of every sign condition over a family of real polynomials defining regular hypersurfaces that intersect transversally. This enables us to show a probabilistic procedure to list all feasible sign conditions. We extend our main algorithm to the case of an...

Testing Sign Conditions on a Multivariate Polynomial and Applications

Let f be a polynomial in Q[X1, . . . , Xn] of degree D. We focus on testing the emptiness and computing at least one point in each connected component of the semi-algebraic set defined by f > 0 (or f < 0 or f 6= 0). To this end, the problem is reduced to computing at least one point in each connected component of a hypersurface defined by f −e = 0 for e ∈ Q positive and small enough. We provide...

Factoring Multivariate Polynomials over Finite Fields

We consider the deterministic complexity of the problem of polynomial factorization over finite fields given a finite field Fq and a polynomial h(x, y) ∈ Fq[x, y] compute the unique factorization of h(x, y) as a product of irreducible polynomials. This problem admits a randomized polynomial-time algorithm and no deterministic polynomial-time algorithm is known. In this chapter, we give a determ...

Factoring Multivariate Polynomials over Large Finite Fields

A simple probabilistic algorithm is presented to find the irreducible factors of a bivariate polynomial over a large finite field. For a polynomial f(x, y) over F of total degree n , our algorithm takes at most 4.89, 2 , n log n log q operations in F to factor f(x , y) completely. This improves a probabilistic factorization algorithm of von zur Gathen and Kaltofen, which takes 0(n log n log q) ...

Factorizing Multivariate Polynomials Over Finite Fields