50 research outputs found
Uniqueness of certain polynomials constant on a line
We study a question with connections to linear algebra, real algebraic
geometry, combinatorics, and complex analysis. Let be a polynomial of
degree with positive coefficients and no negative coefficients, such
that when . A sharp estimate is known. In this paper
we study the for which equality holds. We prove some new results about the
form of these "sharp" polynomials. Using these new results and using two
independent computational methods we give a complete classification of these
polynomials up to . The question is motivated by the problem of
classification of CR maps between spheres in different dimensions.Comment: 20 pages, latex; removed section 10 and address referee suggestions;
accepted to Linear Algebra and its Application
Integer polyhedra for program analysis
Polyhedra are widely used in model checking and abstract interpretation. Polyhedral analysis is effective when the relationships between variables are linear, but suffers from imprecision when it is necessary to take into account the integrality of the represented space. Imprecision also arises when non-linear constraints occur. Moreover, in terms of tractability, even a space defined by linear constraints can become unmanageable owing to the excessive number of inequalities. Thus it is useful to identify those inequalities whose omission has least impact on the represented space. This paper shows how these issues can be addressed in a novel way by growing the integer hull of the space and approximating the number of integral points within a bounded polyhedron
Delta-Decision Procedures for Exists-Forall Problems over the Reals
Solving nonlinear SMT problems over real numbers has wide applications in
robotics and AI. While significant progress is made in solving quantifier-free
SMT formulas in the domain, quantified formulas have been much less
investigated. We propose the first delta-complete algorithm for solving
satisfiability of nonlinear SMT over real numbers with universal quantification
and a wide range of nonlinear functions. Our methods combine ideas from
counterexample-guided synthesis, interval constraint propagation, and local
optimization. In particular, we show how special care is required in handling
the interleaving of numerical and symbolic reasoning to ensure
delta-completeness. In experiments, we show that the proposed algorithms can
handle many new problems beyond the reach of existing SMT solvers