Given a system of polynomial equations and inequations with coefficients in the field of rational numbers, we show how to compute a geometric resolution of the set of common roots of the system over the field of complex numbers. A geometric resolution consists of a primitive element of the algebraic extension defined by the set of roots, its minimal polynomial and the parametrizations of the coordinates. Such a representation of the solutions has a long history which goes back to Leopold Kronecker and has been revisited many times in computer algebra. We introduce a new generation of probabilistic algorithms where all the computations use only univariate or bivariate polynomials. We give a new codification of the set of solutions of a positive dimensional algebraic variety relying on a new global version of Newton’s iterator. Roughly speaking the complexity of our algorithm is polynomial in some kind of degree of the system, in its height, and linear in the complexity of evaluation

We give a unified (“basis free”) framework for the Descartes method for real root isolation of square-free real polynomials. This framework encompasses the usual Descartes ’ rule of sign method for polynomials in the power basis as well as its analog in the Bernstein basis. We then give a new bound on the size of the recursion tree in the Descartes method for polynomials with real coefficients. Applied to polynomials A(X) = P n i=0 aiXi with integer coefficients |ai | < 2 L, this yields a bound of O(n(L + log n)) on the size of recursion trees. We show that this bound is tight for L = Ω(log n), and we use it to derive the best known bit complexity bound for the integer case.

We present algorithmic, complexity and implementation results concerning real root isolation of a polynomial of degree d, with integer coefficients of bit size ≤ τ, using Sturm (-Habicht) sequences and the Bernstein subdivision solver. In particular, we unify and simplify the analysis of both methods and we give an asymptotic complexity bound of eOB(d 4 τ 2). This matches the best known bounds for binary subdivision solvers. Moreover, we generalize this to cover the non square-free polynomials and show that within the same complexity we can also compute the multiplicities of the roots. We also consider algorithms for sign evaluation, comparison of real algebraic numbers and simultaneous inequalities, and we improve the known bounds at least by a factor of d. Finally, we present our C++ implementation in synaps and some preliminary experiments on various data sets.

We propose fast algorithms for computing composed products and composed sums, as well as diamond products of univariate polynomials. These operations correspond to special multivariate resultants, that we compute using power sums of roots of polynomials, by means of their generating series.

We present a new, far simpler family of counter-examples to Kushnirenko’s Conjecture. Along the way, we illustrate a computer-assisted approach to finding sparse polynomial systems with maximally many real roots, thus shedding light on the nature of optimal upper bounds in real fewnomial theory. We use a powerful recent formula for the A-discriminant, and give new bounds on the topology of certain A-discriminant varieties. A consequence of the latter result is a new upper bound on the number of topological types of certain real algebraic sets defined by sparse polynomial equations. 1

We propose fast algorithms for computing composed products and composed sums, as well as diamond products of univariate polynomials. These operations correspond to special resultants, that we compute using power sums of roots of polynomials, by means of their generating series.

Abstract. Let FEASR denote the problem of deciding whether a given system of real polynomial equations has a real root or not. We give a new, nearly tight threshold for when m is large enough to make FEASR be NP-hard for input a single n-variate polynomial with exactly m monomial terms. We also outline a connection between the complexity of FEASR, the topology of A-discriminants, and triangulations of finite point sets. (The A-discriminant contains all known resultants as special cases, and the latter objects are central in algorithmic algebraic geometry.) With this motivation, we then conclude by studying some new cases of A-discriminants whose vanishing can be decided within the polynomial hierarchy. This includes the detection of (a) multiple roots for sparse univariate polynomials and (b) degenerate points on certain “fewnomial ” hypersurfaces. Along the way, we also derive new quantitative bounds on the real zero sets of n-variate (n + 2)-nomials. 1 Introduction and Main Results

We introduce a method to compute the topology of planar algebraic curves. The curve may not be in generic position and may have vertical asymptotes. The algebraic tools are rational univariate representation for zero-dimentional ideals and multiplicities in these ideals. Experiments show the e ciency of our algorithm. 1

Fewnomial theory began with explicit bounds — solely in terms of the number of variables and monomial terms — on the number of real roots of systems of polynomial equations. Here we take the next logical step of investigating the corresponding existence problem: Let FEASR denote the problem of deciding whether a given system of multivariate polynomial equations with integer coefficients has a real root or not. We describe a phase-transition for when m is large enough to make FEASR be NP-hard, when restricted to inputs consisting of a single n-variate polynomial with exactly m monomial terms: polynomial-time for m≤n + 2 (for any fixed n) and NPhardness for m ≥ n + n ε (for n varying and any fixed ε> 0). Because of important connections between FEASR and A-discriminants, we then study some new families of A-discriminants whose signs can be decided within polynomial-time. (A-discriminants contain all known resultants as special cases, and the latter objects are central in algorithmic algebraic geometry.) Baker’s Theorem from diophantine approximation arises as a key tool. Along the way, we also derive new quantitative bounds on the real zero

Let f be a degree D univariate polynomial with real coefficients and at most 3 monomial terms. We show that all the roots of f in any closed interval of length R can be approximated within an accuracy of # using just O(log D log log R # +log 2 D) arithmetic steps, i.e., the arithmetic complexity is polylogarithmic in the degree of the underlying complex variety. In particular, of independent interest is an algebraic sub-routine of our algorithm which counts the roots in any given interval using just O(log 2 D) arithmetic steps. The best previous arithmetic complexity upper bounds for these respective solving and counting problems were O(D log 5 D log log R # ) and O(D log 2 D log log D).