Semi-algebraic decision complexity, the real spectrum, and degree

Complexity of Semi-algebraic Proofs

It is a known approach to translate propositional formulas into systems of polynomial inequalities and consider proof systems for the latter. The well-studied proof systems of this type are the Cutting Plane proof system (CP) utilizing linear inequalities and the Lovász– Schrijver calculi (LS) utilizing quadratic inequalities. We introduce generalizations LS of LS that operate on polynomial ine...

Multi-degree Bounds on the Betti Numbers of Real Varieties and Semi-algebraic Sets and Applications

We prove new bounds on the Betti numbers of real varieties and semi-algebraic sets that have a more refined dependence on the degrees of the polynomials defining them than results known before. Our method also unifies several different types of results under a single framework, such as bounds depending on the total degrees, on multi-degrees, as well as in the case of quadratic and partially qua...

Real Algebraic Numbers: Complexity Analysis and Experimentation

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 e OB(d τ). This matches the best known bounds fo...

Computational Real Algebraic Geometry 3 THE DECISION

Comparing Real Algebraic Numbers of Small Degree

We study polynomials of degree up to 4 over the rationals or a computable real subfield. Our motivation comes from the need to evaluate predicates in nonlinear computational geometry efficiently and exactly. We show a new method to compare real algebraic numbers by precomputing generalized Sturm sequences, thus avoiding iterative methods; the method, moreover handles all degenerate cases. Our f...