Semialgebraic set

In mathematics, a semialgebraic set is a finite union of sets defined by polynomial equalities and polynomial inequalities. A semialgebraic function is a function with a semialgebraic graph. Such sets and functions are mainly studied in real algebraic geometry which is the appropriate framework for algebraic geometry over the real numbers. Let be a real closed field. (For example could be the field of real numbers .) A subset of is a semialgebraic set if it is a finite union of sets defined by polynomial equalities of the form and of sets defined by polynomial inequalities of the form Similarly to algebraic subvarieties, finite unions and intersections of semialgebraic sets are still semialgebraic sets. Furthermore, unlike subvarieties, the complement of a semialgebraic set is again semialgebraic. Finally, and most importantly, the Tarski–Seidenberg theorem says that they are also closed under the projection operation: in other words a semialgebraic set projected onto a linear subspace yields another semialgebraic set (as is the case for quantifier elimination). These properties together mean that semialgebraic sets form an o-minimal structure on R. A semialgebraic set (or function) is said to be defined over a subring A of R if there is some description as in the definition, where the polynomials can be chosen to have coefficients in A. On a dense open subset of the semialgebraic set S, it is (locally) a submanifold. One can define the dimension of S to be the largest dimension at points at which it is a submanifold. It is not hard to see that a semialgebraic set lies inside an algebraic subvariety of the same dimension.

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Convergence Results For Projected Line-Search Methods On Varieties Of Low-Rank Matrices Via Lojasiewicz Inequality

André Uschmajew

The aim of this paper is to derive convergence results for projected line-search methods on the real-algebraic variety M-

Siam Publications2015

Higman ideal, stable Hochschild homology and Auslander-Reiten conjecture

Yuming Liu

Let A and B be two finite dimensional algebras over an algebraically closed field, related to each other by a stable equivalence of Morita type. We prove that A and B have the same number of isomorphism classes of simple modules if and only if their 0-degr ...

Springer-Verlag2012

Real algebraic geometry

In mathematics, real algebraic geometry is the sub-branch of algebraic geometry studying real algebraic sets, i.e. real-number solutions to algebraic equations with real-number coefficients, and mappings between them (in particular real polynomial mappings). Semialgebraic geometry is the study of semialgebraic sets, i.e. real-number solutions to algebraic inequalities with-real number coefficients, and mappings between them. The most natural mappings between semialgebraic sets are semialgebraic mappings, i.

Quantifier elimination

Quantifier elimination is a concept of simplification used in mathematical logic, model theory, and theoretical computer science. Informally, a quantified statement " such that " can be viewed as a question "When is there an such that ?", and the statement without quantifiers can be viewed as the answer to that question. One way of classifying formulas is by the amount of quantification. Formulas with less depth of quantifier alternation are thought of as being simpler, with the quantifier-free formulas as the simplest.

Real closed field

In mathematics, a real closed field is a field F that has the same first-order properties as the field of real numbers. Some examples are the field of real numbers, the field of real algebraic numbers, and the field of hyperreal numbers. A real closed field is a field F in which any of the following equivalent conditions is true: F is elementarily equivalent to the real numbers. In other words, it has the same first-order properties as the reals: any sentence in the first-order language of fields is true in F if and only if it is true in the reals.

Irreducible Algebraic Sets MATH-328: Algebraic geometry I - Curves

Explores irreducible algebraic sets and their unique decomposition into components, with a focus on prime ideals and plane subsets classification.

Projective Space and Algebraic Sets MATH-328: Algebraic geometry I - Curves

Introduces projective space and algebraic sets, discussing homogeneous coordinates, ideals, and generators.

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Covers radical ideals, Nullstellensatz, primary ideals, and algebraic sets.