A Decision Procedure for Univariate Polynomial Systems Based on Root Counting and Interval Subdivision

This paper presents a formally verified decision procedure for determinining the satisfiability of a system of univariate polynomial relations over the real line. The procedure combines a root counting function, based on Sturm’s theorem, with an interval subdivision algorithm. Given a system of polynomial relations over the same variable, the decision procedure progressively subdivides the real interval into smaller intervals. The subdivision continues until the satisfiability of the system can be determined on each subinterval using Sturm’s theorem on a subset of the system’s polynomials. The decision procedure has been formally verified in the Prototype Verification System (PVS). In PVS, the decision procedure is specified as a computable Boolean function on a deep embedding of polynomial relations. This function is used to define a proof producing strategy for automatically proving existential and universal statements on polynomial systems. The soundness of the strategy solely depends on the internal logic of PVS

Cohomology Jumping Loci and the Relative Malcev Completion

Two standard invariants used to study the fundamental group of the complement X of a hyperplane arrangement are the Malcev completion of its fundamental group G and the cohomology groups of X with coefficients in rank one local systems. In this thesis, we develop a tool that unifies these two approaches. This tool is the Malcev completion S_p of G relative to a homomorphism p from G into (C^*)^N. The relative completion S_p is a prosolvable group that generalizes the classical Malcev completion; when p is the trivial representation, S_p is the Malcev completion of G. The group S_p is tightly controlled by the cohomology groups H^1(X,L_{p^k}) with coefficients in the irreducible local systems L_{p^k} associated to the representation p.The pronilpotent Lie algebra u_p of the prounipotent radical U_p of S_p has been described by Hain. If p is the trivial representation, then u_p is the holonomy Lie algebra, which is well-known to be quadratically presented. In contrast, we show that when X is the complement of the braid arrangement in complex two-space, there are infinitely many representations p from G into (C^*)^2 for which u_p is not quadratically presented.We show that if Y is a subtorus of the character torus T containing the trivial character, then S_p is combinatorially determined for general p in Y. We do not know whether S_p is always combinatorially determined. If S_p is combinatorially determined for all characters p of G, then the characteristic varieties of the arrangement X are combinatorially determined.When Y is an irreducible subvariety of T^N, we examine the behavior of S_p as p varies in Y. We define an affine group scheme S_Y over Y such that if Y = {p}, then S_Y is the relative Malcev completion S_p. For each p in Y, there is a canonical homomorphism of affine group schemes from S_p into the affine group scheme which is the restriction of S_Y to p. This is often an isomorphism. For example, if there exists p in Y whose image is Zariski dense in G_m^N, then this homomorphism is an isomorphism for general p in Y.Dissertatio