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Verifying nonlinear real formulas via sums of squares
 Theorem Proving in Higher Order Logics, TPHOLs 2007, volume 4732 of Lect. Notes in Comp. Sci
, 2007
"... Abstract. Techniques based on sums of squares appear promising as a general approach to the universal theory of reals with addition and multiplication, i.e. verifying Boolean combinations of equations and inequalities. A particularly attractive feature is that suitable ‘sum of squares ’ certificates ..."
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Abstract. Techniques based on sums of squares appear promising as a general approach to the universal theory of reals with addition and multiplication, i.e. verifying Boolean combinations of equations and inequalities. A particularly attractive feature is that suitable ‘sum of squares ’ certificates can be found by sophisticated numerical methods such as semidefinite programming, yet the actual verification of the resulting proof is straightforward even in a highly foundational theorem prover. We will describe our experience with an implementation in HOL Light, noting some successes as well as difficulties. We also describe a new approach to the univariate case that can handle some otherwise difficult examples. 1 Verifying nonlinear formulas over the reals Over the real numbers, there are algorithms that can in principle perform quantifier elimination from arbitrary firstorder formulas built up using addition, multiplication and the usual equality and inequality predicates. A classic example of such a quantifier elimination equivalence is the criterion for a quadratic equation to have a real root: ∀a b c. (∃x. ax 2 + bx + c = 0) ⇔ a = 0 ∧ (b = 0 ⇒ c = 0) ∨ a � = 0 ∧ b 2 ≥ 4ac
An effective version of Pólya's theorem on positive definite forms
, 1995
"... Given a real homogeneous polynomial F , strictly positive in the nonnegative orthant, P'olya's theorem says that for a sufficiently large exponent p the coefficients of F (x1 ; . . . ; xn ) \Delta (x1 + \Delta \Delta \Delta +xn ) p are strictly positive. The smallest such p will be call ..."
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Given a real homogeneous polynomial F , strictly positive in the nonnegative orthant, P'olya's theorem says that for a sufficiently large exponent p the coefficients of F (x1 ; . . . ; xn ) \Delta (x1 + \Delta \Delta \Delta +xn ) p are strictly positive. The smallest such p will be called the P'olya exponent of F . We present a new proof for P'olya's result, which allows us to obtain an explicit upper bound on the P'olya exponent when F has rational coefficients. An algorithm to obtain reasonably good bounds for specific instances is also derived. P'olya's theorem has appeared before in constructive solutions of Hilbert's 17th problem for positive definite forms [4]. We also present a different procedure to do this kind of construction. 1 Introduction In 1928 G. P'olya [7] proved the following theorem (see also [5]): Theorem 1.1 (P'olya) Let F (x 1 ; . . . ; x n ) be a real homogeneous polynomial which is positive in x i 0, P x i ? 0. Then, for a sufficiently large integer p, ...
Nullstellensatz and Positivestellensatz from cutelimination
"... We give here an effective proof of Hilbert's nullstellensatz and ..."
Nullstellensatz and Positivstellensatz from cutelimination
"... We give in this article an effective proof of Hilbert's nullstellensatz and KrivineStengle's positivstellensatz using the cut elimination theorem for sequent calculus. The proof is very similar to the current techniques in constructive algebraic geometry by Henri Lombardi, but seems more ..."
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We give in this article an effective proof of Hilbert's nullstellensatz and KrivineStengle's positivstellensatz using the cut elimination theorem for sequent calculus. The proof is very similar to the current techniques in constructive algebraic geometry by Henri Lombardi, but seems more modular. In the case of the positivstellensatz, we think we prove a more general result than the original one, thanks to a new notion of justification of positiveness: PBDD ( polynomial binary decision digram). It allows both to recover KrivineStengle's justification, but also another one which seems to require lower degree. We apply the same techniques to the nullstellensatz for differentially closed field and show that the proof is almost unchanged. Remark: here we do not provide bound, but an effective algorithm, implemented in OCaml, to build the wanted algebraic equality. Nevertheless, we discuss how bounds could probably be obtained. We also do not deal effectively with the axiom of real closure, but we do deal with the axiom of algebraic closure and inverse in the case of the nullstellensatz. However, in both case, those axioms may be eliminated using standard model theory. 1