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Géométrie algèbrique --- Géométrie --- Actions de groupes (mathématiques). --- Problèmes et exercices. --- Etude et enseignement (supérieur). --- Géométrie algèbrique --- Géométrie --- Actions de groupes (mathématiques). --- Problèmes et exercices. --- Etude et enseignement (supérieur). --- Géométrie algébrique --- Actions de groupes (mathématiques)

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Algebraic Geometry is the study of systems of polynomial equations in one or more variables, asking such questions as: Does the system have finitely many solutions, and if so how can one find them? And if there are infinitely many solutions, how can they be described and manipulated? The solutions of a system of polynomial equations form a geometric object called a variety; the corresponding algebraic object is an ideal. There is a close relationship between ideals and varieties which reveals the intimate link between algebra and geometry. Written at a level appropriate to undergraduates, this book covers such topics as the Hilbert Basis Theorem, the Nullstellensatz, invariant theory, projective geometry, and dimension theory. The algorithms to answer questions such as those posed above are an important part of algebraic geometry. This book bases its discussion of algorithms on a generalization of the division algorithm for polynomials in one variable that was only discovered in the 1960's. Although the algorithmic roots of algebraic geometry are old, the computational aspects were neglected earlier in this century. This has changed in recent years, and new algorithms, coupled with the power of fast computers, have let to some interesting applications, for example in robotics and in geometric theorem proving. In preparing a new edition of Ideals, Varieties and Algorithms the authors present an improved proof of the Buchberger Criterion as well as a proof of Bezout's Theorem. Appendix C contains a new section on Axiom and an update about Maple , Mathematica and REDUCE.

Geometry, Algebraic --- Commutative algebra --- Data processing --- Algebraic geometry --- Algèbres commutatives --- Géométrie algébrique --- Algèbres commutatives --- Géométrie algébrique --- Computer. Automation --- Informatique --- Mathematical logic. --- Mathematical Logic and Foundations. --- Algebra of logic --- Logic, Universal --- Mathematical logic --- Symbolic and mathematical logic --- Symbolic logic --- Mathematics --- Algebra, Abstract --- Metamathematics --- Set theory --- Syllogism --- Geometry, Algebraic - Data processing --- Commutative algebra - Data processing --- Géométrie algébrique. --- Algorithms --- Algorithmes. --- Algèbres commutatives.

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Real analytic sets in Euclidean space (Le. , sets defined locally at each point of Euclidean space by the vanishing of an analytic function) were first investigated in the 1950's by H. Cartan [Car], H. Whitney [WI-3], F. Bruhat [W-B] and others. Their approach was to derive information about real analytic sets from properties of their complexifications. After some basic geometrical and topological facts were established, however, the study of real analytic sets stagnated. This contrasted the rapid develop­ ment of complex analytic geometry which followed the groundbreaking work of the early 1950's. Certain pathologies in the real case contributed to this failure to progress. For example, the closure of -or the connected components of-a constructible set (Le. , a locally finite union of differ­ ences of real analytic sets) need not be constructible (e. g. , R - {O} and 3 2 2 { (x, y, z) E R : x = zy2, x + y2 -=I- O}, respectively). Responding to this in the 1960's, R. Thorn [Thl], S. Lojasiewicz [LI,2] and others undertook the study of a larger class of sets, the semianalytic sets, which are the sets defined locally at each point of Euclidean space by a finite number of ana­ lytic function equalities and inequalities. They established that semianalytic sets admit Whitney stratifications and triangulations, and using these tools they clarified the local topological structure of these sets. For example, they showed that the closure and the connected components of a semianalytic set are semianalytic.

Differential geometry. Global analysis --- Semialgebraic sets --- Semianalytic sets --- Topology. --- Algebraic geometry. --- Algebraic topology. --- Mathematical logic. --- Geometry. --- Algebraic Geometry. --- Algebraic Topology. --- Mathematical Logic and Foundations. --- Mathematics --- Euclid's Elements --- Algebra of logic --- Logic, Universal --- Mathematical logic --- Symbolic and mathematical logic --- Symbolic logic --- Algebra, Abstract --- Metamathematics --- Set theory --- Syllogism --- Topology --- Algebraic geometry --- Geometry --- Analysis situs --- Position analysis --- Rubber-sheet geometry --- Polyhedra --- Algebras, Linear --- Semianalytic sets. --- Semialgebraic sets. --- Geometry, Algebraic --- Semi-analytic sets --- Géométrie algébrique --- Géométrie algébrique --- Espaces analytiques