Giới thiệu về Đại số Phi Tuyến: Resultant, Discriminant và Ứng dụng bởi V. Morozov

ITEP/TH-35/06 Introduction to Non-Linear Algebra V.Morozov ITEP, Moscow, Russia ABSTRACT Concise introduction to a relatively new subject of non-linear algebra: literal extension of text-book linear algebra to the case of non-linear equations and maps. This powerful science is based on the notions of discriminant (hyperdeterminant) and resultant, which today can be effectively studied both analytically and by modern computer facilities. The paper is mostly focused on resultants of non-linear maps. First steps are described in direction of Mandelbrot-set theory, which is direct extension of the eigenvalue problem from linear algebra, and is related by renormalization group ideas to the theory of phase transitions and dualities.

arXiv:hep-th/0609022 v2 5 Sep 2006 Contents 1 Introduction 3 1.1 Formulation of the problem .2 Comparison of linear and non-linear algebra .3 Quantities, associated with tensors of different types .1 A word of caution .4 Solutions to poly-linear and non-linear equations .2 Non-homogeneous equations .2 Non-linear equations .1 Homogeneous non-linear equations .2 Solution of systems of non-homogeneous equations: generalized Craemer rule. 19 3 Evaluation of resultants and their properties 20 3.1 Summary of resultant theory .1 Tensors, possessing a resultant: generalization of square matrices .2 Definition of the resultant: generalization of condition det A = 0 for solvability of system of homogeneous linear 3.3 Degree of the resultant: generalization of dn|1 = degA (det A) = n for matrices. composition: generalization of  det AB  =Qdet A det B for determinants 21 j n j 3.5 Resultant for diagonal maps: generalization of det diag aj = j=1 aj for matrices .6 Resultant for matrix-like maps: a more interesting generalization of det diag ajj = nj=1 ajj for matrices 22 P Q σ(i) 3.7 Additive decomposition: generalization of det A = σ (−)σ i Ai for determinants .8 Evaluation of resultants .2 Iterated resultants and solvability of systems of non-linear equations .1 Definition of iterated resultant R̃n|s {A} .3 On the origin of extra factors in R̃ .5 An example of cubic equation .6 Iterated resultant depends on symplicial structure .3 Resultants and Koszul complexes [4]-[8] .2 Linear maps (the case of s1 = .3 A pair of polynomials (the case of n = 2) .4 A triple of polynomials (the case of n = 3). Explicit expression for determinant of exact complex.

Formulation through ǫ-tensors .8 Not only Koszul and not only complexes .4 Resultants and diagram representation of tensor algebra .3 Rectangular tensors and linear maps .4 Generalized Vieta formula for solutions of non-homogeneous equations .5 Coinciding solutions of non-homogeneous equations: generalized discriminantal varieties. 45 4 Discriminants of polylinear forms 45 4.1 Tensors and polylinear forms .3 Degree of discriminant Qr .4 Discriminant as an k=1 SL(n Qr k ) invariant .5 Diagram technique for the k=1 SL(nk ) invariants .6 Symmetric, diagonal and other specific tensors .7 Invariants from group averages .8 Relation to resultants .2 Discrminants and resultants: Degeneracy condition .1 Direct solution to discriminantal constraints .2 Degeneracy condition in terms of det T̂ .5 Degeneracy of the product .6 An example of consistency between (4.3 Discriminants and complexes .1 Koshul complexes, associated with poly-linear and symmetric functions .2 Reductions of Koshul complex for poly-linear tensor .3 Reduced complex for generic bilinear n × n tensor: discriminant is determinant of the square matrix 55 4.4 Complex for generic symmetric discriminant .2 Discriminant through paths .3 Discriminants from diagrams. 58 5 Examples of resultants and discriminants 59 5.1 The case of rank r = 1 (vectors) .2 The case of rank r = 2 (matrices) .4 Symmetric hypercubic tensors 2×r and polynomials of a single variable .7) and its analogues in the n = 2 case .1 Direct evaluation of Z(T ) .2 Gaussian integrations: specifics of cases n = 2 and r = 2 .3 Alternative partition functions .4 Pure tensor-algebra (combinatorial) partition functions .com 6 Eigenspaces, eigenvalues and resultants 87 6.1 From linear to non-linear case .2 Eigenstate (fixed point) problem and characteristic equation .2 Number of eigenvectors cn|s as compared to the dimension Mn|s of the space of symmetric functions 89 6.8) of characteristic equation: example of diagonal map .8) of characteristic equation: non-diagonal example for n|s = 2|2 .5 Numerical examples of decomposition (6.3 Eigenvalue representation of non-linear map .2 Eigenvalue representation of Plukker coordinates .3 Examples for diagonal maps .5 Map from its eigenvectors: the case of n|s = 2|2 .6 Appropriately normalized eigenvectors and elimination of Λ-parameters .4 Eigenvector problem and unit operators .1 Relation between Rn|s2 (λs+1 |A◦2 ) and Rn|s (λ|A) .2 Unit maps and exponential of maps: non-linear counterpart of algebra ↔ group relation .3 Examples of exponential maps .1 Exponential maps for n|s = 2|2 .2 Examples of exponential maps for 2|s .3 Examples of exponential maps for n|s = 3|2 .2 Number of solutions .3 Index of projective map .2 Dynamical systems theory .1 Bifurcations of maps, Julia and Mandelbrot sets .2 The universal Mandelbrot set .3 Relation between discrete and continuous dynamics: iterated maps, RG-like equations and effective actions112 8.1 Basic example: matrix case, n|r = n|2 .2 Basic example: polynomial case, n|r = 2|r .3 Integrals of polylinear forms .4 Multiplicativity of integral discriminants .5 Cayley 2 × 2 × 2 hyperdeterminant as an example of coincidence between integral and algebraic discriminants119 8.5 Differential equations and functional integrals .1 Formulation of the problem Linear algebra [1] is one of the foundations of modern natural science: wherever we are interested in calculations, from engineering to string theory, we use linear equations, quadratic forms, matrices, linear maps and their cohomologies. There is a widespread feeling that the non-linear world is very different, and it is usually studied as a sophisticated phenomenon of interpolation between different approximately-linear regimes.

In [2] we already explained that this feeling can be wrong: non-linear world, with all its seeming complexity including ”chaotic structures” like Julia and Mandelbrot sets, allows clear and accurate description in terms of ordinary algebraic geometry. In this paper we extend this analysis to generic multidimensional situation and show that non-linear phenomena are direct generalizations of the linear ones, without any approximations. The thing is that the theory of generic tensors and associated multi-linear functions and non-linear maps can be built literally repeating everything what is done with matrices (tensors of rank 2), as summarized in the table in sec. It appears that the only essential difference is the 3 www.com lack of ”obvious” canonical representations (like sum of squares for quadratic forms or Jordan cells for linear maps): one can not immediately choose between different possibilities.1 All other ingredients of linear algebra and, most important, its main ”special function” – determinant – have direct (moreover, literal) counterparts in non-linear case.

Of course, this kind of ideas is hardly new [4]-[12], actually, they can be considered as one of constituents of the string program [13]. However, for mysterious reasons – given significance of non-linear phenomena – the field remains practically untouched and extremely poor in formulas. In this paper we make one more attempt to convince scientists and scholars (physicists, mathematicians and engineers) that non-linear algebra is as good and as powerfull as the linear one, and from this perspective we”ll see one day that the non-linear world is as simple and transparent as the linear one. This world is much bigger and more diverse: there are more ”phases”, more ”cohomologies”, more ”reshufflings” and ”bifurcations”, but they all are just the same as in linear situation and adequate mathematical formalism does exist and is essentially the same as in linear algebra.

One of the possible explanations for delay in the formulation of non-linear algebra is the lack of adequate computer facilities even in the close past. As we explain below, not all the calculations are easily done ”by bare hands” even in the simplest cases. Writing down explicit expression for the simplest non-trivial resultant R3|2 – a non-linear generalization of the usual determinant – is similar to writing 12! terms of explicit expression for determinant of a 12 × 12 matrix: both tedious and useless. What we need to know are properties of the quantity and possibility to evaluate it in a particular practical situation.

For example, for particular cubic form 31 ax3 + 13 by 3 + 13 cz 3 + 2ǫxyz the resultant is given by a simple and practical expression: R3|2 = abc(abc + 8ǫ3 )3. Similarly, any other particular case can be handled with modern computer facilities, like MAPLE or Mathematica. A number of results below are based on computer experiments. At the heart of our approach to quantitative non-linear algebra are special functions – discriminants and resultants – generalizations of determinant in linear algebra.

Sometime, when they (rarely) appear in modern literature [7, 14] these functions are called hyperdeterminants, but since we define them in terms of consistency (existence of common solutions) of systems of non-linear equations, we prefer to use ”discrminantal” terminology [9, 10, 11]. At least at the present stage of developement the use of such terminology is adequate in one more respect. One of effective ways to evaluate discriminants and resultants exploits the fact that they appear as certain irreducible factors in various auxiliary problems, and constructive definitions express them through iterative application of two operations: taking an ordinary resultant of two functions of a single variable and taking an irreducible factor. The first operation is constructively defined (and well computerized) for polynomials (see s.10 of [2] for directions of generalization to arbitrary functions), and in this paper we restrict consideration to non-linear, but polynomial equations, i.

to the theory of tensors of finite rank or – in homogeneous coordinates – of functions and maps between projective spaces P n−1. The second operation – extraction of irreducible factor, denoted irf (.) in what follows, – is very clear conceptually and very convenient for pure-science considerations, but it is a typical N P problem from calculational point of view. Moreover, when we write, say, D = irf (R), this means that D is a divisor of R, but actually in most cases we mean more: that it is the divisor, the somehow distinguished irreducible factor in R (in some cases D can be divisor of R, where R can be obtained in slightly different ways, for example by different sequences of iterations, – then D is a common divisor of all such R). Therefore, at least for practical purposes, it is important to look for ”direct” definitions/representations of discriminants and resultants (e.

like row and column decompositions of ordinary determinants), even if aestetically disappealing, they are practically usefull, and – no less important – they provide concrete definition of irf operattion. Such representations were suggested already in XIX century [4, 5], and the idea was to associate with original non-linear system some linear-algebra problem (typically, a set of maps between some vector spaces), which degenerates simulteneously with the original system. Then discriminantal space acquires a linear-algebra representation and can be studied by the methods of homological algebra [15]. First steps along these lines are described in [7], see also s.4 in the present paper.

Another option is Feynman-style diagram technique [16, 17], capturing the structure of convolutions in tensor algebra with only two kinds of invariant tensors involved: the unity δji and totally antisymmetric ǫi1. Diagrams provide all kinds of invariants, made from original tensor, of which discriminant or resultant is just one. Unfortunately, enumeration and classification of diagrams is somewhat tedious and adequate technique needs to be found for calculation of appropriate generating functions. Distinction between discriminants and resultants in this paper refers to two essentially different types of objects: functions (analogues of quadratic forms in linear algebra) and maps.

From tensorial point of view this is distinction between pure covariant tensors and those with both contravariant and covariant indices (we mostly consider the case of a single contravariant index). The difference is two-fold.