Semialgebraic Subset Research Articles

Analytic and differentiable functions vanishing on an algebraic set

Let U U be an open semi-algebraic subset of R n {{\mathbf {R}}^n} and let X X be a closed analytic subset of U U which is also a semi-algebraic set (e.g., U = R n U = {{\mathbf {R}}^n} and X X is an algebraic subset of R n {{\mathbf {R}}^n} ). It is proved that the ideal of analytic functions on U U vanishing on X X is finitely generated provided that the set X X is coherent. The ideal of infinitely differentiable functions on U U vanishing on X X is finitely generated if and only if the set X X is coherent.

Nash wings and real prime divisors

Let V be an affine variety defined over an arbitrary real closed field R with affine coordinate ring A = R[V] and function field K = R(V). Then open semialgebraic subsets of the regular points of V(R) define what we call oeometric preorders on K. The study of affine semialgebraic geometry corresponds to the study of commutative algebra in K in the presence of such a preorder. The theorem which motivated this paper is

An Approach to Simultaneous System Design, I. Semialgebraic Geometric Methods

This paper introduces semialgebraic parameterization as an approach to analyze simultaneous stabilization and pole placement problems. Rational families of plants of a given McMillan degree, that are simultaneously stabilizable by a fixed family of compensators, are parameterized. For a discrete family of plants, the parameterization problem reduces to the simultaneous stabilization or the pole placement problem of a r-tuple of multi input multi output plants by a nonswitching compensator. It is shown that by removing a semialgebraic subset of a proper algebraic set, the “space of plants” can be decomposed into components that are either simultaneously stabilizable or simultaneously unstabilizable. Under special cases, explicit parameterization of the semialgebraic set is obtained. Finally a necessary condition for the simultaneous stabilization of single input or single output plants is obtained.

Real proper morphisms

Introduction. In this paper we study proper morphisms between real algebraic varieties in two different settings. In the first paragraph we use the concept of semi-integral extensions J BI[ to define semi-integral morphisms which turn out to be analogous (if we work in the family of semialgebraic subsets) of proper maps. In the second paragraph we use the language of real spectra to give an approach to proper morphisms as it is done classically in schemas over algebraically closed fields. Let us mention that proper morphisms have been also studied by Delfs and Knebusch [6] in their general development of semialgebraic spaces. Also I have just known of some related work by Ramanakoraisina [8] on the same topic.

Embedding semi-algebraic spaces

Let R be any real closed field (which we take to be totally ordered) and let A" denote n-dimensional affine R-space. An affine semi-algebraic set is a subset of some A" which has a finite description in terms of polynomial equalities and inequalitities. A semi-algebraic space is an object obtained by pasting finitely many semi-algebraic sets together along open semi-algebraic subsets and may be thought of as the semi-algebraic version of an integral separated R-scheme of finite type. Semi-algebraic spaces were introduced in [2] by Delfs and Knebusch and continue to be the objects of intensive and fruitful investigation by these authors. The question which we address is "Can every semi-algebraic space be embedded in some affine space?" This question was raised in [2] and the answer is provided by

A Characterization of C ∞ -Sufficient k-Jets

We improve some results of Mather and Arnold and find several necessary and sufficient conditions of sufficiency of k-jets. As a corollary we prove that the set of C?-sufficient k-jets is a semialgebraic subset of the space of k-jets of C' mappings F: (Rn, 0) -+ (R, 0).

A characterization of 𝐶^{∞}-sufficient 𝑘-jets

We improve some results of Mather and Arnold and find several necessary and sufficient conditions of sufficiency of k k -jets. As a corollary we prove that the set of C ∞ {C^\infty } -sufficient k k -jets is a semialgebraic subset of the space of k k -jets of C ∞ {C^\infty } mappings F : ( R n , 0 ) → ( R , 0 ) F:({R^n},0) \to (R,0) .

Smooth functions invariant under the action of a compact lie group

be a compact Lie group acting orthogonally on IX”. By classical theorem of Hilbert ([14], p. 274), the algebra P((w”)G of G-invariant polynomials on Iw” is finitely generated. Let p,, . . . , pk be generators and let p = (p,, . . . , pk) denote the corresponding map from [w” to Dg’. The square of the radius function on Iw” is a proper map, and it is a polynomial in the pi’s Hence p is proper. Since P((w”)’ separates the orbits of G and since the orbit space IF/G is locally compact Hausdorff ([3], p. 38), p induces a homeomorphism p’ of [W”/G with the closed semi-algebraic subset p([w”) of [w” ([l], p. 100). lR”/G can be given a “smooth structure” by defining a function on KY/G to be smooth if it pulls back to a smooth function on R”, and p(Iw”) has a smooth structure defined by restricting the C” functions 6(Rk) on DBk to p([w”). It has been conjectured that p is an isomorphism of [W”/G and p([w”) together with their smooth structures. The conjecture is known in some special cases and it has proved useful in obtaining classification theorems for certain types of smooth group actions ([3], Ch. VI; [4]). Of course, the conjecture is equivalent to THEOREM 1.