An ultrametric field is a field that is locally compact as a metric space with respect to a non-archimedean absolute value. The main topic of this dissertation is to study roots of polynomials over such fields. If we have a univariate polynomial with coefficients in an ultrametric field and non-vanishing discriminant, then there is a bijection between the set of roots of the polynomial and classes of roots of the same polynomial in a finite ring. As a consequence, there is a ball in the polynomial space where all polynomials in it have the same number of roots. If a univariate polynomial satisfies certain generic conditions, then we can efficiently compute the exact number of roots in the field. We do that by using Hensel's lemma and some properties of Newton's polygon. In the multivariate case, if we have a square system of polynomials, we consider the tropical set which is the intersection of the tropical varieties of its polynomials. The tropical set contains the set of valuations of the roots, and for every point in the tropical set, there is a corresponding system of lower polynomials. If the system satisfies some generic conditions, then for each point w in the tropical set the number of roots of valuation w equals the number roots of valuation w of the lower system. The last result enables us to compute the exact number of roots of a polynomial system where the tropical set is finite and the lower system consists of binomials. This algorithmic method can be performed in polynomial-time if we fix the number of variables. We conclude the dissertation with a discussion of the feasibility problem. We consider the problem of the p-adic feasibility of polynomials with integral coefficients with the prime number p as a part of the input. We prove this problem can be solved in nondeterministic polynomial-time. Furthermore, we show that any problem, which can be solved in nondeterministic polynomial-time, can be reduced to this feasibility problem in randomized polynomial-time.

In this work we present two algebraic certificates for Budan's theorem. Budan's theorem claims the following. Let R be an ordered field, f in R[X] of degree n and a,b in R with a

Dans ce travail, nous présentons deux certificats algébriques pour le théorème de Budan. Le théorème de Budan s'énonce comme suit : Soit R un corps ordonné, f in R[X] de degré n et a,b in R avec a<b. Alors, le nombre de variations de signe dans la suite (f(b),f'(b),...,f^n(b)) n'est pas supérieur au nombre de variations de signe dans la séquence (f(a),f'(a),...,f^n(a)). Cela nous permet de compter des racines réelles d'une manière similaire au comptage des racines réelles par le théorème de Sturm. (Compter des racines réelles à la Budan est aujourd'hui connu comme Budan-Fourier count. En effet, il compte des racines dites virtuelles qui comprennent les racines réelles.) Un certificat algébrique pour le théoème de Budan est un certain type de preuve qui mène de la négation de l'hypothèse à l'identité algébrique contradictionelle 0>0. L'algorithme pour notre premier certificat est basé sur la preuve historique par Budan, qui utilise uniquement des arguments combinatoires. Il a une complexité exponentielle dans le degré de f. L'algorithme pour le deuxième certificat est basé sur des suites de Taylor mixtes et exhibe une plus petite complexité : Le calcul principal est la résolution d'un système linéaire, ce qui est polynomiale dans le degré de f / In this work we present two algebraic certificates for Budan's theorem. Budan's theorem claims the following. Let R be an ordered field, f in R[X] of degree n and a,b in R with a<b. Then the number of sign changes in the sequence (f(b),f'(b),...,f^n(b)) is not greater than the number of sign changes in the sequence (f(a),f'(a),...,f^n(a)). This enables us to count real roots in a similar way to the real root counting by Sturm's theorem. (Budan's count of real roots is today known as ``Budan-Fourier count'' which, indeed, counts so called virtual roots which comprehend the real roots.) An algebraic certificate for Budan's theorem is a certain kind of proof which leads from the negation of the assumption to the contradictory algebraic identity 0>0. The algorithm for our first certificate is based on the historical proof by Budan which uses only combinatorial arguments. It has a complexity exponential in the degree of f. The algorithm for the second certificate is based on mixed Taylor series and shows a smaller complexity: The main calculation is solving a linear system; this is polynomial in the degree of f.