Algorithms in Real Algebraic Geometry

Abstract

We survey both old and new developments in the theory of algorithms in real algebraic geometry -starting from effective quantifier elimination in the first order theory of reals due to Tarski and Seidenberg, to more recent algorithms for computing topological invariants of semi-algebraic sets. We emphasize throughout the complexity aspects of these algorithms and also discuss the computational hardness of the underlying problems. We also describe some recent results linking the computational hardness of decision problems in the first order theory of the reals, with that of computing certain topological invariants of semi-algebraic sets. Even though we mostly concentrate on exact algorithms, we also discuss some numerical approaches involving semi-definite programming that have gained popularity in recent times. Contents

Application of Sturm Theorem in the global controllability of a class of high dimensional polynomial systems

“…By the real roots isolation algorithm (see [7,12,13]), we obtain two sequences of disjoint intervals {I i }, {I j }. I i and I j contain one root of p 1 (x) and p 2 (x) exactly, respectively.…”

“…In this section, we first introduce Sturm Theorem, which is used for deciding how many real roots a univariate polynomial P (x) ∈ R[x] has in any interval [a, b], where R[x] denotes the univariate polynomial ring in variable x with coefficients in real number field R. Definition 2.1 (see [7,8,11]) If c = {c 1 , c 2 , · · · , c m } is a finite sequence of non-zero elements in R, then we call the number of sign variations in c to be the number of i,…”

“…Furthermore, the length of every interval can be made arbitrarily short if it is necessary. More details about the algorithm please refer to [7,[12][13][14].…”

“…In this paper, we will use the methods and results in real algebraic geometry/mathematics mechanization to investigate the global controllability for a class of higher dimensional polynomial systems and propose a constructive criterion algorithm which only needs the coefficient of the polynomials, namely, we can determine the global controllability in finite steps of arithmetic operations. The analysis is based on a classical result-Sturm Theorem and its modern progress in real algebraic geometry [7,8] and mathematics mechanization theory [9,10] . Sturm Theorem ever played a key role in the proof of Routh-Hurwitz criterion in the stability theory of linear system, especially in the proof of the Routh criterion/algorithm which tells us more information about the eigenvalues [11] .…”

Stability of Disease Free Equilibria in Epidemiological Models

“…One may indeed use a Routh-Hurwitz like criterion to formulate the problem as a first order formula in the language of ordered fields and then resort to real algebra algorithms as a simplification tool, see e.g., [2,5,12,14,16]. This approach has the advantage of being completely algorithmic, but due to the very complex combinatorial nature of simplification of first order formulae it may produce a quite involved condition for some models whereas hand calculations are successful in producing a simple equivalent condition, see e.g., [5].…”