A real (complex) algebraic subset of Rn (respectively, Cn) is defined by the vanishing of finitely many real (respectively, complex) polynomials of n variables. That is,
A = {(x1,...,xn) : P1(x1,...,xn)= ... = Pj(x1,...,xn)= 0 }
Classically their study is at the heart of algebraic geometry. Unions and intersections of finitely many algebraic sets are again algebraic. If one also allows polynomial inequalities, then one obtains the class of semi-algebraic sets. While a given semi-algebraic set may fail to be a manifold, its singular points form a closed “lower-dimensional” semi-algebraic set. One may thus verify inductively that a semi-algebraic set admits a finite partition S={Si} (called a stratification) into semi-algebraic, connected (open) submanifolds of various dimensions. Each individual stratum enjoys various nice properties. For example, a bounded semi-algebraic stratum of dimension k has finite k dimensional Hausdorff measure. A semi-algebraic set admits such a semi-algebraic stratification where the closures of the strata are topological simplices providing a triangulation of the set. A complex algebraic set admits a stratification into complex strata which are topologically K(Π,1)'s, that is having trivial homotopy in degrees > 1. One can also impose some restrictions on the possible fundamental groups Π1(S) of the strata in such special complex stratifications.
In a stratification, the closure in Rn of a given stratum S is the disjoint union of S and various lower dimensional strata. It is important to study how these fit together topologically and metrically. There are topologically trivial stratificiations where the family of all strata S sharing a given stratum T in their closures form, near T, a topologically trivial bundle over T . Examples show that the trivializing maps may not be algebraic or even differentiable, but they may be semi-algebraic, i.e. have semi-algebraic graphs.
Metrically, Whitney established properties concerning the behavior of the tangent planes to a stratum at points approaching a point on the boundary stratum. Lojasiewicz also discovered several interesting inequalities concerning relative distances to the various strata. Orders of contact between adjacent strata at common boundary points is always finite and even rational.
Most of the arguments carry over to semi-analytic sets which are similarly defined using real analytic functions. An exception is the Tarski-Seidenberg Theorem which says that a linear projection of a semi-algebraic set is again semi-algebraic. Since linear projection corresponds to the algebraic process of eliminating variables, this theorem corresponds to a statement in logic concerning the polynomial decidability of polynomially defined statements with variables eliminated. Semi-algebraic sets occur in many other contexts in pure and applied mathematics. For example, in robotics one may be concerned with finding efficient paths in various semi-algebraic constained state spaces, e.g. in the problem of moving furniture through a hallway or up stairs.
Our goal in this course is to prove many of these results using only the classical implicit function theorem and a first course in complex analysis. In the first few weeks we will develop a few needed facts from several complex variables.
Meets: MWF 11-11:50 in Herman Brown 453
Instructor: Robert Hardt, Herman Brown 430; Office hours: 1-2 MWF (and others by appt.),
Email: hardt@rice.edu, Telephone: ext 3280
H. Whitney, Tangents to an Analytic Variety, Ann. Math 81(1965), 496-549.
R. Gunning and H. Rossi, Analytic Functions of Several Complex Variables Prentice-Hall, 1965.
R. Hardt, Semi-algebraic local triviality in semi-algebraic mappings Amer. J. Math102(1980), 291-302.
E. Bierstone and P. Milman, Semi-analytic and Subanalytic Sets, I.H.E.S. Publications 67(1988), 5-42.
This page is maintained by Robert Hardt ( email)
Last edited 8/05/07.