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Meeting Times: Thursday 11:15-12:15
Starting Date: Thursday, September 14
Room: 409 Lind Hall
Weekly
Schedule Tutorial
Notes
Aim of this tutorial: to provide a transition from elementary real algebraic geometry (represented, for example by Cox, Little and O’Shea (2005)) to more advanced topics in this area (represented, for example, by Basu, Pollack and Roy (2003)).
We shall mainly be interested in semi-algebraic sets. These are subsets of real n-space that are defined by a finite number of polynomial equations and polynomial inequalities.
One of the main results in this area is Tarski’s theorem that the elementary theory of algebra and geometry is decidable. (See Tarski(1951).) In particular, whether a semi-algebraic set is non-empty is decidable. It also follows that every subset of real n-space definable by a first order formula is semi-algebraic.
Prerequisites for this tutorial: a basic knowledge of linear algebra and the rudiments of a basic course in algebra through the definitions and basic properties of groups, rings and fields, and in topology through the elementary properties of closed, open, compact and connected sets.
This tutorial should be of interest to students and researchers in mathematics, computer science and engineering
References:
Weekly Schedule 11:15am, Lind Hall 409 (unless specified) |
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Distance matrices
Abstract: A "distance matrix" is a matrix containing the distances, taken pairwise, of a set of points. I shall trace the history of distance matrices during the last 150 years. I shall develop distance matrices from an inner product. I shall present some applications and at least one generalization.
Border bases (continued)
Abstract: We discussed border bases on May 21 and June 7 June 14 and June 21. We shall continue that discussion.
I continue to look at the following books and papers:
Kehrein, A; Kreuzer, M. and Robbiano, L. (2005) "An algebraist's view of border basis", in Dickenstein, A. and Emiris, I.Z. (2005), Solving Polynomial Equations}, Springer.
Mourrain, B. (1999) "A new criterion for normal form algorithms", in Applied algebra, algebraic algorithms and error-correcting codes (Honolulu, HI, 1999), 430-443, Lecture Notes in Computer Science, 1719, Springer.
Stetter, H.J. (2004) Numerical Polynomial Algebra, SIAM.
I plan to mainly follow the last few chapters in the book by Stetter.
As usual, I shall try to minimize prerequisites.
Border bases (continued)
Abstract: We discussed border bases on May 21 and June 7 June 14. We shall continue that discussion.
I continue to look at the following books and papers:
Kehrein, A; Kreuzer, M. and Robbiano, L. (2005) "An algebraist's view of border basis," in Dickenstein, A. and Emiris, I.Z. (2005), Solving Polynomial Equations}, Springer.
Mourrain, B. (1999) "A new criterion for normal form algorithms," in Applied algebra, algebraic algorithms and error-correcting codes (Honolulu, HI, 1999), 430-443, Lecture Notes in Computer Science, 1719, Springer.
Stetter, H.J. (2004) Numerical Polynomial Algebra, SIAM.
I shall continue to mainly follow the paper/chapter by Kehrein, Kreuzer and Robbiano.
As usual, I shall try to minimize prerequisites.
Border bases (continued)
Abstract: We discussed border bases on May 21 and June 7. We shall continue that discussion.
I continue to look at the following books and papers:
Kehrein, A; Kreuzer, M. and Robbiano, L. (2005) "An algebraist's view of border basis", in Dickenstein, A. and Emiris, I.Z. (2005), Solving Polynomial Equations}, Springer.
Mourrain, B. (1999) "A new criterion for normal form algorithms", in Applied algebra, algebraic algorithms and error-correcting codes (Honolulu, HI, 1999), 430-443, Lecture Notes in Computer Science, 1719, Springer.
Stetter, H.J. (2004) Numerical Polynomial Algebra, SIAM.
I shall continue to mainly follow the paper by Kehrein, Kreuzer and Robbiano.
As usual, I shall try to minimize prerequisites.
Border bases (continued)
Abstract: We discussed border bases for a short time on May 21. We shall continue that discussion (after a review since we did not meet last week). I have been looking at the following books and papers:
Kehrein, A; Kreuzer, M. and Robbiano, L. (2005) "An algebraist's view of border basis", in Dickenstein, A. and Emiris, I.Z. (2005), Solving Polynomial Equations}, Springer.
Mourrain, B. (1999) "A new criterion for normal form algorithms", in Applied algebra, algebraic algorithms and error-correcting codes (Honolulu, HI, 1999), 430-443, Lecture Notes in Computer Science, 1719, Springer.
Stetter, H.J. (2004) Numerical Polynomial Algebra, SIAM.
I shall mainly follow the paper by Kehrein, Kreuzer and Robbiano.
As usual, I shall try to minimize prerequisites.
Numerical polynomial algebra
Abstract: See the book "Numerical Polynomial Algebra" by H. J. Stetter, SIAM, 2004.
Sums of squares of polynomials (continued)
Abstract: The value of a multivariate polynomial that is a sum of squares of polynomials is always weakly positive. But there are some multivariate polynomials that are always weakly positive and yet not sums of squares. We shall continue to discuss this matter.
Sums of squares of polynomials
Abstract: The value of a multivariate polynomial that is a sum of squares of polynomials is always weakly positive. But there are some multivariate polynomials that are always weakly positive and yet not sums of squares. We shall discuss this matter.
I shall use the following paper as a basis for our discussion:
"An algorithm for sums of squares of real polynomials" by V. Powers and T. Woermann, J. of Pure and Applied Algebra 127 (1998) .
You can also find a copy of the paper on Thorsten Woermann's web site.
In the paper, Powers and Woermann present "an algorithm to determine if a real polynomial is a sum of squares of polynomials,and an explicit representation if it is a sum of squares. This algorithm uses the fact that a sum of squares representation of a real polynomial corresponds to a real, symmetric, positive semi-definite matrix whose entries satisfy certain liner equations."
We shall discuss a number of applications of this algorithm.
As usual I shall try to minimize prerequisites.
Some applications of real root counting to mechanics
Abstract: This talk is intended to illustrate some of the methods covered last semester in the real algebraic geometry tutorial. I will describe some problems from celestial mechanics which can be reduced to counting real roots of systems of polynomial equations. I will show how to solve some of the simplest ones by using Sturm's algorithm and Hermite root counting and will mention several open problems.
Real and complex varieties: comparisons, contrasts and
examples
html
version of the presentation
Abstract: Properties of a complex variety often can be illustrated by a picture of its real locus. Conversely, we can often understand a real variety by studying the corresponding complex variety. At the same time, there are cases where the properties of real varieties and complex varieties are quite different. This can include basic properties like connectedness, compactness, and local dimension near singular points.
In this talk we will exhibit pictures of some curves and surfaces, that illustrate some of these similarities and differences. These figures will include several types of surfaces in R^{3}.
Many of the surface pictures are interactive, in that they are posted on a webpage, where the viewer — using nothing more than a Java-enabled browser — can rotate the figure continuously by dragging it with the mouse.
Kenneth R. Driessel (Department of Mathematics, Iowa State University)Also recall that a commutative ring with identity is "real" if the only representation of zero as the sum of squares is the trivial one.
Since we did not meet during the last two weeks, I shall include an substantial review as part of our discussion.
I shall mainly follow the material in the chapter "Real Rings" in the book "Positive Polynomials" by Prestel and Delzell.
Our objective will be a proof of an absrtact version of the real Nullstellensatz.
This will be a new topic for us. I shall try to minimize prerequisites.
I shall mainly follow the material in the chapter "Real Rings" in the book "Positive Polynomials" by Prestel and Delzell.
Our objective will be a proof of the real Nullstellensatz.