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. Introduction 4 Formulation of the problem 4 Comparison of linear and non-linear algebra 6 Quantities, associated with tensors of different types 10 A word of caution 10 Tensors 10 Tensor algebra 11 Solutions to poly-linear and non-linear equations 14 Solving equations. Resultants 17 Linear algebra (particular case of s=1) 17 Homogeneous equations 17 Non-homogeneous equations 17 Non-linear equations 18 Homogeneous non-linear equations 18 Solution of systems of non-homogeneous equations: generalized Craemer rule 20 Evaluation of resultants and their properties 21 Summary of resultant theory 21 Tensors, possessing a resultant: generalization of square matrices 21 Definition of the resultant: generalization of condition detA = 0 for solvability of system of homogeneous linear equations 22 Degree of the resultant: generalization of dn|1 = degA (detA) = n for matrices 22 Multiplicativity w.r.t. composition: generalization of detAB = detA detB for determinants 22 Resultant for diagonal maps: generalization of det(to1.5.diag ajj)to1.5. = j=1n ajj for matrices 23 Resultant for matrix-like maps: a more interesting generalization of det(to1.5.diag ajj )to1.5. = j=1n ajj for matrices 23 Additive decomposition: generalization of detA = (-)i Ai(i) for determinants 24 Evaluation of resultants 25 Iterated resultants and solvability of systems of non-linear equations 25 Definition of iterated resultant n|s{A} 25 Linear equations 25 On the origin of extra factors in 27 Quadratic equations 28 An example of cubic equation 28 Iterated resultant depends on symplicial structure 29 Resultants and Koszul complexes Cay-AG 29 Koszul complex. I. Definitions 29 Linear maps (the case of s1=...=sn=1) 30 A pair of polynomials (the case of n=2) 31 A triple of polynomials (the case of n=3) 31 Koszul complex. II. Explicit expression for determinant of exact complex 32 Koszul complex. III. Bicomplex structure 35 Koszul complex. IV. Formulation through -tensors 35 Not only Koszul and not only complexes 37 Resultants and diagram representation of tensor algebra 39 Tensor algebras T(A) and T(T), generated by AiI and T GMS 39 Operators 39 Rectangular tensors and linear maps 40 Generalized Vieta formula for solutions of non-homogeneous equations 41 Coinciding solutions of non-homogeneous equations: generalized discriminantal varieties 46 Discriminants of polylinear forms 46 Definitions 47 Tensors and polylinear forms 47 Discriminantal tensors 47 Degree of discriminant 47 Discriminant as an k=1r SL(nk) invariant 48 Diagram technique for the k=1r SL(nk) invariants 49 Symmetric, diagonal and other specific tensors 49 Invariants from group averages 50 Relation to resultants 50 Discrminants and resultants: Degeneracy condition 51 Direct solution to discriminantal constraints 51 Degeneracy condition in terms of det 51 Constraint on P[z] 52 Example 52 Degeneracy of the product 53 An example of consistency between (4.17) and (4.19) 53 Discriminants and complexes 53 Koshul complexes, associated with poly-linear and symmetric functions 53 Reductions of Koshul complex for poly-linear tensor 54 Reduced complex for generic bilinear nn tensor: discriminant is determinant of the square matrix 56 Complex for generic symmetric discriminant 57 Other representations 58 Iterated discriminant 58 Discriminant through paths 59 Discriminants from diagrams 59 Examples of resultants and discriminants 60 The case of rank r=1 (vectors) 60 The case of rank r=2 (matrices) 61 The 222 case (Cayley hyperdeterminant Cay) 64 Symmetric hypercubic tensors 2r and polynomials of a single variable 67 Generalities 67 The n|r = 2|2 case 68 The n|r = 2|3 case 71 The n|r = 2|4 case 72 Functional integral (1.7) and its analogues in the n=2 case 75 Direct evaluation of Z(T) 75 Gaussian integrations: specifics of cases n=2 and r=2 79 Alternative partition functions 80 Pure tensor-algebra (combinatorial) partition functions 83 Tensorial exponent 87 Oriented contraction 88 Generating operation ("exponent") 88 Beyond n=2 88 Eigenspaces, eigenvalues and resultants 88 From linear to non-linear case 88 Eigenstate (fixed point) problem and characteristic equation 89 Generalities 89 Number of eigenvectors cn|s as compared to the dimension Mn|s of the space of symmetric functions 90 Decomposition (6.8) of characteristic equation: example of diagonal map 91 Decomposition (6.8) of characteristic equation: non-diagonal example for n|s = 2|2 94 Numerical examples of decomposition (6.8) for n>2 95 Eigenvalue representation of non-linear map 95 Generalities 95 Eigenvalue representation of Plukker coordinates 96 Examples for diagonal maps 96 The map f(x) = x2 + c: 98 Map from its eigenvectors: the case of n|s = 2|2 99 Appropriately normalized eigenvectors and elimination of -parameters 100 Eigenvector problem and unit operators 102 Iterated maps 102 Relation between Rn|s2(s+1|A2) and Rn|s(|A) 103 Unit maps and exponential of maps: non-linear counterpart of algebra group relation 105 Examples of exponential maps 106 Exponential maps for n|s=2|2 106 Examples of exponential maps for 2|s 107 Examples of exponential maps for n|s=3|2 108 Potential applications 108 Solving equations 109 Craemer rule 109 Number of solutions 109 Index of projective map 111 Dynamical systems theory 112 Bifurcations of maps, Julia and Mandelbrot sets 112 The universal Mandelbrot set 113 Relation between discrete and continuous dynamics: iterated maps, RG-like equations and effective actions 113 Jacobian problem 117 Taking integrals 117 Basic example: matrix case, n|r = n|2 118 Basic example: polynomial case, n|r = 2|r 118 Integrals of polylinear forms 118 Multiplicativity of integral discriminants 119 Cayley 222 hyperdeterminant as an example of coincidence between integral and algebraic discriminants 120 Differential equations and functional integrals 120 Acknowledgements 120 "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."--Jacket Presents the domain of consistent non-linear counterparts for different basic objects and tools of linear algebra. This volume develops an adequate calculus for solving non-linear algebraic and differential equations. It uses a diagram technique and incorporates various illustrations