Introduction to non-linear algebra

This 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.

Preface	p. vii
1 Introduction	p. 1
1.1 Comparison of linear and non-linear algebra	p. 5
1.2 Quantities, associated with tensors of different types	p. 12
1.2.1 A word of caution	p. 12
1.2.2 Tensors	p. 14
1.2.3 Tensor algebra	p. 16
1.2.4 Solutions to poly-linear and non-linear equations	p. 21
2 Solving Equations. Resultants	p. 29
2.1 Linear algebra (particular case of s = 1)	p. 29
2.1.1 Homogeneous equations	p. 29
2.1.2 Non-homogeneous equations	p. 30
2.2 Non-linear equations	p. 31
2.2.1 Homogeneous non-linear equations	p. 31
2.2.2 Solution of systems of iron-homogeneous equations: Generalized Cramer rule	p. 34
3 Evaluation of Resultants and Their Properties	p. 39
3.1 Summary of resultant theory	p. 39
3.1.1 Tensors, possessing a resultant: Generalization of square matrices	p. 39
3.1.2 Definition of the resultant: Generalization of condition det A = 0 for solvability of system of homogeneous linear equations	p. 40
3.1.3 Degree of the resultant: Generalization of d[subscript n/2] = deg[subscript A] (det A) = n for matrices	p. 40
3.1.4 Multiplicativity w.r.t. composition: Generalization of det AB = det A det B for determinants	p. 41
3.1.5 Resultant for diagonal maps: Generalization of det [Characters not reproducible] for matrices	p. 42
3.1.6 Resultant for matrix-like maps: A more interesting generalization of det [Characters not reproducible] for matrices	p. 42
3.1.7 Additive decomposition: Generalization of det A = [Characters not reproducible] for determinants	p. 44
3.1.8 Evaluation of resultants	p. 45
3.2 Iterated resultants and solvability of systems of non-linear equations	p. 46
3.2.1 Definition of iterated resultant R[subscript n/s] {{A}}	p. 46
3.2.2 Linear equations	p. 47
3.2.3 On the origin of extra factors in R	p. 49
3.2.4 Quadratic equations	p. 50
3.2.5 An example of cubic equation	p. 51
3.2.6 More examples of 1-parametric deformations	p. 52
3.2.7 Iterated resultant depends on simplicial structure	p. 52
3.3 Resultants and Koszul complexes	p. 52
3.3.1 Koszul complex. I. Definitions	p. 53
3.3.2 Linear maps (the case of s[subscript 1] = ... = s[subscript n] = 1)	p. 55
3.3.3 A pair of polynomials (the case of n = 2)	p. 56
3.3.4 A triple of polynomials (the case of n = 3)	p. 57
3.3.5 Koszul complex. II. Explicit expression for determinant of exact complex	p. 59
3.3.6 Koszul complex. III. Bicomplex structure	p. 62
3.3.7 Koszul complex. IV. Formulation through [epsilon]-tensors	p. 64
3.3.8 Not only Koszul and not only complexes	p. 66
3.4 Resultants and diagram representation of tensor algebra	p. 69
3.4.1 Tensor algebras T(A) and T(T), generated by [Characters not reproducible] and T	p. 70
3.4.2 Operators	p. 71
3.4.3 Rectangular tensors and linear maps	p. 73
3.4.4 Generalized Vieta formula for solutions of non-homogeneous equations	p. 74
3.4.5 Coinciding solutions of non-homogeneous equations: Generalized discriminantal varieties	p. 80
4 Discriminants of Polylinear Forms	p. 85
4.1 Definitions	p. 85
4.1.1 Tensors and polylinear forms	p. 85
4.1.2 Discriminantal tensors	p. 86
4.1.3 Degree of discriminant	p. 86
4.1.4 Discriminant as an [Characters not reproducible] invariant	p. 88
4.1.5 Diagram technique for the [Characters not reproducible] invariants	p. 89
4.1.6 Symmetric, diagonal and other specific tensors	p. 90
4.1.7 Invariants from group averages	p. 92
4.1.8 Relation to resultants	p. 92
4.2 Discriminants and resultants: Degeneracy condition	p. 94
4.2.1 Direct solution to discriminantal constraints	p. 94
4.2.2 Degeneracy condition in terms of det T	p. 95
4.2.3 Constraint on P[z]	p. 96
4.2.4 Example	p. 96
4.2.5 Degeneracy of the product	p. 97
4.2.6 An example of consistency between (4.18) and (4.22)	p. 98
4.3 Discriminants and complexes	p. 99
4.3.1 Koszul complexes, associated with poly-linear and symmetric functions	p. 99
4.3.2 Reductions of Koszul complex for poly-linear tensor	p. 101
4.3.3 Reduced complex for generic bilinear n x n tensor: Discriminant is determinant of the square matrix	p. 105
4.3.4 Complex for generic symmetric discriminant	p. 107
4.4 Other representations	p. 107
4.4.1 Iterated discriminant	p. 107
4.4.2 Discriminant through paths	p. 109
4.4.3 Discriminants from diagrams	p. 110
5 Examples of Resultants and Discriminants	p. 111
5.1 The case of rank r = 1 (vectors)	p. 111
5.2 The case of rank r = 2 (matrices)	p. 113
5.3 The 2 x 2 x 2 case (Cayley hyperdeterminant)	p. 119
5.4 Symmetric hypercubic tensors 2[superscript xr] and polynomials of a single variable	p. 125
5.4.1 Generalities	p. 125
5.4.2 The n/r = 2/2 case	p. 132
5.4.3 The n/r = 2/3 case	p. 137
5.4.4 The n/r = 2/4 case	p. 138
5.5 Functional integral (1.7) and its analogues in the n = 2 case	p. 143
5.5.1 Direct evaluation of Z(T)	p. 143
5.5.2 Gaussian integrations: Specifics of cases n = 2 and r = 2	p. 150
5.5.3 Alternative partition functions	p. 151
5.5.4 Pure tensor-algebra (combinatorial) partition functions	p. 154
5.6 Tensorial exponent	p. 160
5.6.1 Oriented contraction	p. 160
5.6.2 Generating operation ("exponent")	p. 160
5.7 Beyond n = 2	p. 161
5.7.1 D[subscript 3/3], D[subscript 3/4] and D[subscript 4/3] through determinants	p. 161
5.7.2 Generalization: Example of non-Koszul description of generic symmetric discriminants	p. 164
6 Eigenspaces, Eigenvalues and Resultants	p. 173
6.1 From linear to non-linear case	p. 173
6.2 Eigenstate (fixed point) problem and characteristic equation	p. 174
6.2.1 Generalities	p. 174
6.2.2 Number of eigenvectors c[subscript n/s] as compared to the dimension M[subscript n/s] of the space of symmetric functions	p. 176
6.2.3 Decomposition (6.8) of characteristic equation: Example of diagonal map	p. 178
6.2.4 Decomposition (6.8) of characteristic equation: Non-diagonal example for n/s = 2/2	p. 181
6.2.5 Numerical examples of decomposition (6.8) for n > 2	p. 183
6.3 Eigenvalue representation of non-linear map	p. 183
6.3.1 Generalities	p. 183
6.3.2 Eigenvalue representation of Plucker coordinates	p. 185
6.3.3 Examples for diagonal maps	p. 185
6.3.4 The map f(x) = x[superscript 2] + c	p. 188
6.3.5 Map from its eigenvectors: The case of n/s = 2/2	p. 189
6.3.6 Appropriately normalized eigenvectors and elimination of A-parameters	p. 192
6.4 Eigenvector problem and unit operators	p. 194
7 Iterated Maps	p. 197
7.1 Relation between R[subscript n/s[superscript 2]] ([lambda][superscript s+1]/A[superscript o2]) and R[subscript n/s] ([lambda]/A)	p. 198
7.2 Unit maps and exponential of maps: Non-linear counterpart of algebra [leftrightarrow] group relation	p. 201
7.3 Examples of exponential maps	p. 203
7.3.1 Exponential maps for n/s = 2/2	p. 203
7.3.2 Examples of exponential maps for 2/s	p. 205
7.3.3 Examples of exponential maps for n/s = 3/2	p. 205
8 Potential Applications	p. 209
8.1 Solving equations	p. 209
8.1.1 Cramer rule	p. 209
8.1.2 Number of solutions	p. 211
8.1.3 Index of projective map	p. 212
8.1.4 Perturbative (iterative) solutions	p. 215
8.2 Dynamical systems theory	p. 221
8.2.1 Bifurcations of maps, Julia and Mandelbrot sets	p. 221
8.2.2 The universal Mandelbrot set	p. 223
8.2.3 Relation between discrete and continuous dynamics: Iterated maps, RG-like equations and effective actions	p. 225
8.3 Jacobian problem	p. 231
8.4 Taking integrals	p. 231
8.4.1 Basic example: Matrix case, n/r = n/2	p. 232
8.4.2 Basic example: Polynomial case, n/r = 2/r	p. 232
8.4.3 Integrals of polylinear forms	p. 233
8.4.4 Multiplicativity of integral discriminants	p. 234
8.4.5 Cayley 2 x 2 x 2 hyperdeterminant as an example of coincidence between integral and algebraic discriminants	p. 236
8.5 Differential equations and functional integrals	p. 236
8.6 Renormalization and Bogolubov's recursion formula	p. 237
9 Appendix: Discriminant D[subscript 3/3](S)	p. 241
Bibliography	p. 267

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