Pablo Solernó

Bounds for traces in complete intersections and degrees in the Nullstellensatz

In this paper we obtain an effective Nullstellensatz using quantitative considerations of the classical duality theory in complete intersections. Letk be an infinite perfect field and let f1,...,f n−r∈k[X1,...,Xn] be a regular sequence with d:=maxj deg fj. Denote byA the polynomial ringk [X1,..., Xr] and byB the factor ring k[X1,...,Xn]/(f1,...,fnr); assume that the canonical morphism A→B is injective and integral and that the Jacobian determinantΔ with respect to the variables Xr+1,...,Xn is not a zero divisor inB. Let finally σ∈B*:=HomA(B, A) be the generator of B* associated to the regular sequence.We show that for each polynomialf the inequality deg σ(¯f) ≦dnr(δ+1) holds (¯fdenotes the class off inB andδ is an upper bound for (n−r)d and degf). For the usual trace associated to the (free) extensionA ↪B we obtain a somewhat more precise bound: deg Tr(¯f) ≦ dnr degf. From these bounds and Bertinis theorem we deduce an elementary proof of the following effective Nullstellensatz: let f1,..., fs be polynomials in k[X1,...,Xn] with degrees bounded by a constant d≧2; then 1 ∈(f1,..., fs) if and only if there exist polynomials p1,..., ps∈k[X1,..., Xn] with degrees bounded by 4n(d+ 1)n such that 1=Σipifi. in the particular cases when the characteristic of the base fieldk is zero ord=2 the sharper bound 4ndn is obtained.

Description of the connected components of a semialgebraic set in single exponential time

This paper is devoted to the following result: letR be a real closed field and letS be a semialgebraic subset ofRn defined by a boolean combination of polynomial inequalities. LetD be the sum of the degrees of the polynomials involved. Then it is possible to find algorithmically a description of the semialgebraically connected components ofS in sequential timeDno(1) and parallel time (n logD)o(1) This implies that the problem of finding the connected components of a semialgebraic set can be solved in P-SPACE.

Deformation Techniques for Sparse Systems

We exhibit a probabilistic symbolic algorithm for solving zero-dimensional sparse systems. Our algorithm combines a symbolic homotopy procedure, based on a flat deformation of a certain morphism of affine varieties, with the polyhedral deformation of Huber and Sturmfels. The complexity of our algorithm is cubic in the size of the combinatorial structure of the input system. This size is mainly represented by the cardinality and mixed volume of Newton polytopes of the input polynomials and an arithmetic analogue of the mixed volume associated to the deformations under consideration.

Effective Łojasiewicz inequalities in semialgebraic geometry

AbstractThe main result of this paper can be stated as follows: letV ⊂ ℝn be a compact semialgebraic set given by a boolean combination of inequalities involving only polynomials whose number and degrees are bounded by someD > 1. LetF, G∈∝[X1,⋯, Xn] be polynomials with degF, degG ≦ D inducing onV continuous semialgebraic functionsf, g:V→R. Assume that the zeros off are contained in the zeros ofg. Then the following effective Łojasiewicz inequality is true: there exists an universal constantc1∈ℕ and a positive constantc2∈∝ (depending onV, f,g) such that

On the Theoretical and Practical Complexity of the Existential Theory of Reals

|g(x)|^{D^{c_1 .n} } \leqq c_2 \cdot |f(x)|

On Intrinsic Bounds in the Nullstellensatz

for allx∈V. This result is generalized to arbitrary given compact semialgebraic setsV and arbitrary continuous functionsf,g:V → ∝. An effective global Łojasiewicz inequality on the minimal distance of solutions of polynomial inequalities systems and an effective Finiteness Theorem (with admissible complexity bounds) for open and closed semialgebraic sets are derived.

Lower complexity bounds for interpolation algorithms

AbstractLetF be a unimodularr×s matrix with entries beingn-variate polynomials over an infinite fieldK. Denote by deg(F) the maximum of the degrees of the entries ofF and letd=1+deg(F). We describe an algorithm which computes a unimodulars×s matrixM with deg(M)=(rd)O(n) such thatFM=[Ir,O], where [Ir,O] denotes ther×s matrix obtained by adding to ther×r unit matrixIrs−r zero columns.We present the algorithm as an arithmetic network with inputs fromK, and we count field operations and comparisons as unit cost.The sequential complexity of our algorithm amounts to