Introduction To Non-linear AlgebraThis unique text presents the new domain of consistent non-linear counterparts for all basic objects and tools of linear algebra, and develops an adequate calculus for solving non-linear algebraic and differential equations. It reveals the non-linear algebraic activity as an essentially wider and diverse field with its own original methods, of which the linear one is a special restricted case.This volume contains a detailed and comprehensive description of basic objects and fundamental techniques arising from the theory of non-linear equations, which constitute the scope of what should be called non-linear algebra. The objects of non-linear algebra are presented in parallel with the corresponding linear ones, followed by an exposition of specific non-linear properties treated with the use of classical (such as the Koszul complex) and original new tools. This volume extensively uses a new diagram technique and is enriched with a variety of illustrations throughout the text. Thus, most of the material is new and is clearly exposed, starting from the elementary level. With the scope of its perspective applications spreading from general algebra to mathematical physics, it will interest a broad audience of physicists; mathematicians, as well as advanced undergraduate and graduate students. |
Contents
| 1 | |
2 Solving Equations Resultants | 29 |
3 Evaluation of Resultants and Their Properties | 39 |
4 Discriminants of Polylinear Forms | 85 |
5 Examples of Resultants and Discriminants | 111 |
6 Eigenspaces Eigenvalues and Resultants | 173 |
7 Iterated Maps | 197 |
8 Potential Applications | 209 |
Discriminant D33S | 241 |
| 267 | |
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Common terms and phrases
actually Ai(z analogue arbitrary associated choice cn|s codimension coefficients coincide covariant Cramer rule decomposition defined degenerate denote detA determinant diagonal map dimension discriminantal constraint Dn|r eigenvalues eigenvectors example exponential expression homogeneous coordinates homogeneous equations homogeneous polynomials homogeneous variables hypercubic intentionally left blank invariant inverse irreducible factor iterated resultants Koszul complex linear algebra linear maps Mandelbrot set Mn|s multi-index non-homogeneous system non-linear algebra non-linear maps non-trivial non-vanishing solution ǫ-tensors obtained operator orbits parameters particular partition functions permutations Plucker poly-linear polynomials of degree problem projective space quadratic rectangular reduced relations representation Rn|s roots simplest SL(n sort-mixing sort-respecting square matrix structure group subspace symmetric tensor tensor algebra tensors of rank theory tion transformations vanishes vector spaces vertices Vieta formula zero