Physical objects and constraints may be modeled by polynomial equations and inequalities. For this reason algebraic geometry, the study of solutions to systems of polynomial equations, is a tool for scientists and engineers. Moreover, relations between concepts arising in science and engineering are often described by polynomials. Whatever their source, once polynomials enter the picture, notions from algebraic geometry—its theoretical base, trove of classical examples, and modern computational tools—may all be brought to bear on the problem at hand. As a part of applied mathematics, algebraic geometry has two faces. One is an expanding list of recurring techniques and examples which are common to many applications, and the other consists of topics from the applied sciences which involve polynomials. Linking these two aspects are algorithms and software for algebraic geometry.

Algebraic Geometry

(here, > is the Maple prompt). Once the Groebner package is loaded, you can perform the division algorithm, compute Groebner bases, and carry out a variety of other commands described below. In Maple, a monomial ordering is called a monomial order. The monomial orderings lex, grlex, and grevlex from Chapter 2 are easy to use in Maple. Lex order is called plex (for “pure lexicographic”), grlex order is called grlex, and grevlex order is called tdeg (for “total degree”). Be careful not to confuse tdeg with grlex. Since a monomial order depends also on how the variables are ordered, Maple needs to know both the monomial order you want (plex, grlex or tdeg) and a list of variables. For example, to tell Maple to use lex order with variables x > y > z, you would need to input plex(x,y,z). The Groebner package also knows some elimination orders, as defined in Exercise 5 of Chapter 3, §1. To eliminate the first k variables from x1, . . . , xn, one can use the monomial order lexdeg([x 1,. . .,x k],[x {k+1},. . . ,x n]) (remember that Maple encloses a list inside brackets [. . .]). This order is the elimination order of Bayer and Stillman described in Exercise 6 of Chapter 3, §1. The Maple documentation for the Groebner package also describes how to use certain weighted orders, and we will explain below how matrix orders give us many more monomial orderings. The most commonly used commands in the Groebner package are NormalForm, for doing the division algorithm, and Basis, for computing a Groebner basis. NormalForm has the following syntax:

/pdf/ideals-varieties-and-algorithms-5wny8dsa96.pdf

Ideals, Varieties, and Algorithms

Background: Polynomial Systems Homotopy Continuation Projective Spaces Probability One Polynomials of One Variable Other Methods Isolated Solutions: Coefficient-Parameter Homotopy Polynomial Structures Case Studies Endpoint Estimation Checking Positive Dimensional Solution Sets: Some Concepts from Algebraic Geometry Basic Numerical Algebraic Geometry Cascading Through Embedded Systems The Numerical Irreducible Decomposition Intersection of Algebraic Sets.

The numerical solution of systems of polynomials arising in engineering and science

Tensors are ubiquitous in the sciences. The geometry of tensors is both a powerful tool for extracting information from data sets, and a beautiful subject in its own right. This book has three intended uses: a classroom textbook, a reference work for researchers in the sciences, and an account of classical and modern results in (aspects of) the theory that will be of interest to researchers in geometry. For classroom use, there is a modern introduction to multilinear algebra and to the geometry and representation theory needed to study tensors, including a large number of exercises. For researchers in the sciences, there is information on tensors in table format for easy reference and a summary of the state of the art in elementary language. This is the first book containing many classical results regarding tensors. Particular applications treated in the book include the complexity of matrix multiplication, $\mathbf{P}$ versus $\mathbf{NP}$, signal processing, phylogenetics, and algebraic statistics. For geometers, there is material on secant varieties, $G$-varieties, spaces with finitely many orbits and how these objects arise in applications, discussions of numerous open questions in geometry arising in applications, and expositions of advanced topics such as the proof of the Alexander-Hirschowitz theorem and of the Weyman-Kempf method for computing syzygies.

Tensors: Geometry and Applications

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Algebraic Geometry

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References

Ideals, Varieties, and Algorithms

The numerical solution of systems of polynomials arising in engineering and science

Tensors: Geometry and Applications