On differential rational invariants of patches with

On differential rational invariants of patches with respect to motion groups Ural Bekbaev Citation: AIP Conference Proceedings 1660, 090048 (2015); doi: 10.1063/1.4926637 View online: http://dx.doi.org/10.1063/1.4926637 View Table of Contents: http://scitation.aip.org/content/aip/proceeding/aipcp/1660?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Integral invariants for curves in 2D with respect to projective groups AIP Conf. Proc. 1605, 524 (2014); 10.1063/1.4887643 Fundamental solutions for partial differential equations with reflection group invariance J. Math. Phys. 36, 4324 (1995); 10.1063/1.530964 On left invariant Brownian motions and heat kernels of nilpotent Lie groups J. Math. Phys. 31, 278 (1990); 10.1063/1.528911 Group invariance in nonlinear motion of rods and strings J. Acoust. Soc. Am. 76, 1169 (1984); 10.1121/1.391409 Geometric Approach to Invariance Groups and Solution of Partial Differential Systems J. Math. Phys. 12, 653 (1971); 10.1063/1.1665631

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 1.9.65.115 On: Fri, 01 Jan 2016 10:17:31

On Differential Rational Invariants of Patches with Respect to Motion Groups Ural Bekbaev Department of Science in Engineering, Faculty of Engineering, International Islamic University Malaysia, 50728 Kuala Lumpur, Malaysia Abstract. This paper can be considered as a research on Algebraic Differential Geometry. It is about differential rational invariants of subgroups of the Affine group over the constant fields of partial differential fields (characteristic zero). The obtained results can be formulated in terms of Differential Geometry as follows: 1. For any motion group represented by a subgroup H of the Affine group it is shown that systems of generators of a field of H-invariant (not differential) rational functions can be used to construct systems of generators for the differential field of H-invariant differential rational functions of parameterized surface (patch). 2. For some classic motion groups H the generating systems of the field of H-invariant differential functions are presented. 3. For motion groups, including all classical subgroups of the Affine group, separating systems of invariants, uniqueness and existence theorems are offered. Keywords: Affine group; patch; differential rational unction; differential invariant. PACS: 02.40.Dr, 02.40.Hw, 03.65.Fd

INTRODUCTION Let ݊ǡ ݉ be any fixed natural numbers, ‫ ܪ‬be any subgroup of the general linear group ‫ܮܩ‬ሺ݊ǡ ܴሻ, where ܴ stands for the field of real numbers. Consider, for example, its identity representation in ܴ௡ : ሺܿǡ ݄ሻ հ ݄ܿ, where ܿ ൌ ሺܿଵ ǡ ܿଶ ǡ Ǥ Ǥ Ǥ ǡ ܿ௡ ሻ ‫ܴ א‬௡ - is a row vector, ݄ ‫ܪ א‬. In the invariant theory the descriptions of ܴሾ࢞ሿு -the algebra of ‫ܪ‬-invariant polynomials in ࢞, and ܴሺ࢞ሻு -the field of ‫ܪ‬-invariant rational functions in ࢞ are considered as one of the main problems if not the most important, where ࢞ ൌ ሺ‫ݔ‬ଵ ǡ ‫ݔ‬ଶ ǡ Ǥ Ǥ Ǥ ǡ ‫ݔ‬௡ ሻ and ‫ݔ‬ଵ ǡ ‫ݔ‬ଶ ǡ Ǥ Ǥ Ǥ ǡ ‫ݔ‬௡ are algebraic independent variables over ܴ. For many, in particular for classic, groups the corresponding descriptions are known [1,2]. At the same time in geometry the description of invariants of ݉-parametric surfaces (patches) ࢞ ൌ ሺ‫ݔ‬ଵ ሺ‫ݐ‬ଵ ǡ Ǥ Ǥ Ǥ ǡ ‫ݐ‬௠ ሻǡ ‫ݔ‬ଶ ሺ‫ݐ‬ଵ ǡ Ǥ Ǥ Ǥ ǡ ‫ݐ‬௠ ሻǡ ‫ ڮ‬ǡ ‫ݔ‬௡ ሺ‫ݐ‬ଵ ǡ Ǥ Ǥ Ǥ ǡ ‫ݐ‬௠ ሻሻ in ܴ௡ with respect to the motion group ‫ ܪ‬is also one of the important problems. In this case consideration of ܴሼ࢞Ǣ ߲ሽு - the algebra of ‫ܪ‬-invariant ߲-differential polynomials as an important object is not convenient as far as even for finite ‫ ܪ‬it, as a ߲-differential algebra డ డ డ ሻ. For over ܴ, may have no finite system of differential generators over ܴ, where ߲ stands for ሺ ǡ ǡ Ǥ Ǥ Ǥ ǡ డ௧భ డ௧మ

డ௧೘

the corresponding field ܴ‫࢞ۃ‬Ǣ ߲ۧு - the field of the ‫ܪ‬-invariant ߲-differential rational functions in ࢞ there is no such problem. It has a finite system of differential generators over ܴ and ܴሺ࢞ሻு ‫࢞ۃܴ ؿ‬Ǣ ߲ۧு . Therefore by algebraic point of view for the parameterized surfaces it is an important object to describe. So in these cases one deals with description of the following algebraic structures. 1. In algebraic case the field of ‫ܪ‬-invariant rational functions ܴሺ࢞ሻு , which are relatively well studied. 2. In algebraic differential geometric case the ߲-differential field of ‫ܪ‬-invariant differential rational functions ܴ‫࢞ۃ‬Ǣ ߲ۧு . The main result of this paper states that for any subgroup of the Affine group the second case can be reduced to the first kind case. In reality this paper deals with this question in a more general form. But to show the importance of the considered problems and results we motivate them by the use of differential geometry of patches-where initially they come from. In the finite subgroup and ݉ ൌ ͳ (ordinary differential) cases more accurate and general results can be obtained. For the finite subgroup ‫ ܪ‬case in [3] it is shown that every system of generators of ܴሺ࢞ሻு generates ܴ‫࢞ۃ‬Ǣ ߲ۧு as a ߲-differential field as well. For the ordinary differential case one can see [4]. The main frame of this paper has been highlighted in [5]. In our algebraic approach we use some notions and results of Differential Algebra which can be found in [6, 7].

DIFFERENTIAL RATIONAL ࡴ-INVARIANTS OF PATCHES To set up our investigation problems let us consider the following typical geometric problem. Let ݊, ݉ be natural numbers, ‫ ܪ‬be a subgroup of the affine group ‫ܮܩ‬ሺ݊ǡ ܴሻ ‫ܴ ڈ‬௡ , ࡮ ‫ܴ ؿ‬௠ be open unit ball and ࢛ǣ ࡮ ՜ ܴ௡ be a parameterized surface (patch), where ܴ is the field of real numbers, ࢛ is considered to be infinitely smooth, written in row form. Of course one can consider ࢛ as a vector field over ࡮ as well. But in future we deal with ࢛ as a patch. International Conference on Mathematics, Engineering and Industrial Applications 2014 (ICoMEIA 2014) AIP Conf. Proc. 1660, 090048-1–090048-11; doi: 10.1063/1.4926637 © 2015 AIP Publishing LLC 978-0-7354-1304-7/$30.00

090048-1 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 1.9.65.115 On: Fri, 01 Jan 2016 10:17:31

Definition 2.1: A function ݂ డ ሺ࢛ሺ࢚ሻሻ of ࢛ሺ࢚ሻ ൌ ሺ‫ݑ‬ଵ ሺ࢚ሻǡ Ǥ Ǥ Ǥ ǡ ‫ݑ‬௡ ሺ࢚ሻሻ and its finite number of derivatives డ డ is said to be invariant (more exactly, ‫ܪ‬- invariant) if the equality relative to ǡ Ǥ Ǥ Ǥ ǡ డ௧భ

డ௧೘

݂ డ ሺ࢛ሺ࢚ሻሻ ൌ ݂ డ ሺ࢛ሺ࢚ሻ݄ ൅ ݄଴ ሻ holds for any ሺ݄ǡ ݄଴ ሻ ‫ ܪ א‬and ࢚ ‫࡮ א‬. To represent the above considered problem in a different way let ࢚ run ࡮ and ‫ ܨ‬ൌ ‫ ܥ‬ஶ ሺ࡮ሻ be the differential డ డ ǡǤǤǤǡ . Every infinitely smooth surface ring of infinitely smooth functions relative to differential operators డ௧భ

డ௧೘

࢛ǣ ࡮ ՜ ܴ௡ can be regarded as an element of differential module ሺ‫ ܨ‬௡ Ǣ ߲ଵ ǡ ߲ଶ ǡ Ǥ Ǥ Ǥ ǡ ߲௠ ሻ, with the coordinate-wise డ on elements of ‫ ܨ‬௡ Ǥ If elements of this module are written in row form the above action of ߲௜ ൌ డ௧೔

transformation looks like ࢛ ൌ ሺ‫ݑ‬ଵ ǡ Ǥ Ǥ Ǥ ǡ ‫ݑ‬௡ ሻ հ ࢛݄ ൅ ݄଴ , where ሺ݄ǡ ݄଴ ሻ ‫ܪ א‬. Therefore the following algebraic analogue of the above problem is natural. Let ሺ‫ܨ‬Ǣ ߲ሻ be any differential field, where ߲ stands for the commuting system of differential operators ߲ଵ ǡ Ǥ Ǥ Ǥ ǡ ߲௠ of ‫ܨ‬, ‫ ܥ‬ൌ ሼܽ ‫ܨ א‬ǣ ߲௜ ܽ ൌ Ͳ݅ ൌ ͳǡ Ǥ Ǥ Ǥ ǡ ݉ሽ be its constant field, ‫ ܪ‬be a subgroup of ‫ܮܩ‬ሺ݊ǡ ‫ܥ‬ሻ ‫ ܥ ڈ‬௡ . Let in future ‫ݔ‬ଵ ǡ Ǥ Ǥ Ǥ ǡ ‫ݔ‬௡ be ߲-differential algebraic independent variables over ‫ ܨ‬and ࢞ stand for the row vector with coordinates ‫ݔ‬ଵ ǡ Ǥ Ǥ Ǥ ǡ ‫ݔ‬௡ . We use the following notations : ‫ܥ‬ሾ࢞ሿ - the ring of polynomials in ‫ݔ‬ଵ ǡ Ǥ Ǥ Ǥ ǡ ‫ݔ‬௡ (over ‫)ܥ‬, ‫ܥ‬ሺ࢞ሻ- the field of rational functions in ࢞, ‫ܥ‬ሼ࢞ǡ ߲ሽ -the ring of ߲-differential polynomial functions in ࢞ and ‫࢞ۃܥ‬ǡ ߲ۧ-is the field of ߲-differential rational functions in ࢞ over ‫ܥ‬. Definition 2.2: An element ݂ డ ‫࢞ۃܥ א ۧ࢞ۃ‬Ǣ ߲ۧ is said to be ‫ܪ‬- invariant if the equality ݂ డ ‫ ݄࢞ۃ‬൅ ݄଴ ۧ ൌ ݂ డ ‫ۧ࢞ۃ‬ holds for any ሺ݄ǡ ݄଴ ሻ ‫ܪ א‬. Let ‫࢞ۃܥ‬Ǣ ߲ۧு stand for the field of all such ‫ܪ‬- invariant elements of ‫࢞ۃܥ‬Ǣ ߲ۧ. In future all fields are assumed to be of characteristic zero. Let ሺ‫ܨ‬ǡ ߲ሻ be a differential field. The following result provides a method to describe the differential field ‫࢞ۃܥ‬ǡ ߲ۧு over ‫ ܥ‬by the use of ordinary( not differential) rational invariants of ‫ܪ‬. Let ߙ ଵ ǡ ߙ ଶ ǡ Ǥ Ǥ Ǥ ǡ ߙ ௡ be any different nonzero ݉-tuples with nonnegative integer entries. ೙

భ

మ

೙

Theorem 2.1: The equality ‫࢞ۃܥ‬ǡ ߲ۧீ௅ሺ௡ǡ஼ሻ‫ڈ‬஼ ሺ࢞ǡ ߲ ఈ ࢞ǡ ߲ ఈ ࢞ǡ Ǥ Ǥ Ǥ ǡ ߲ ఈ ࢞ሻ ൌ ‫࢞ۃܥ‬ǡ ߲ۧis valid and moreover the భ మ ೙ ೙ system consisting of components of ࢞ǡ ߲ ఈ ࢞ǡ ߲ ఈ ࢞ǡ Ǥ Ǥ Ǥ ǡ ߲ ఈ ࢞ is algebraic independent over ‫࢞ۃܥ‬ǡ ߲ۧீ௅ሺ௡ǡ஼ሻ‫ן‬஼ . Proof: For any nonzero ݉-tuple ߙ consider the following linear differential equation in ‫ݕ‬: భ

߲ ఈ ‫ݔ‬ଵ ተ ఈ೙‫ڭ‬ ߲ ‫ݔ‬ଵ ߲ ఈ ‫ݔ‬ଵ

భ

߲ ఈ ‫ݔ‬ଶ ‫ڭ‬ ೙ ߲ ఈ ‫ݔ‬ଶ ߲ ఈ ‫ݔ‬ଶ

‫ڮ‬ ‫ڭ‬ ‫ڮ‬ ‫ڮ‬

భ

߲ ఈ ‫ݔ‬௡ ‫ڭ‬ ೙ ߲ ఈ ‫ݔ‬௡ ߲ ఈ ‫ݔ‬௡

భ

భ ߲ఈ ‫ݕ‬ ߲ ఈ ‫ݔ‬ଵ ‫ ڭ‬ተ ቮ ‫ڭ‬ ೙ ߲ ఈ ‫ ݕ‬ఈ೙ ߲ ‫ݔ‬ଵ ఈ ߲ ‫ݕ‬

భ

߲ ఈ ‫ݔ‬ଶ ‫ڭ‬ ೙ ߲ ఈ ‫ݔ‬ଶ

‫ڮ‬ ‫ڭ‬ ‫ڮ‬

భ

߲ ఈ ‫ݔ‬௡ ‫ ڭ‬ቮ ೙ ߲ ఈ ‫ݔ‬௡

ିଵ

ൌͲ

೙

All coefficients of this differential equation are in ‫࢞ۃܥ‬ǡ ߲ۧீ௅ሺ௡ǡ஼ሻ‫ڈ‬஼ , the coefficient at ߲ ఈ ‫ ݕ‬is 1 and ‫ ݕ‬ൌ ‫ݔ‬௜ is a solution whenever ݅ ൌ ͳǡʹǡ Ǥ Ǥ Ǥ ǡ ݊. Therefore ೙

భ

మ

೙

‫࢞ۃܥ‬ǡ ߲ۧீ௅ሺ௡ǡ஼ሻ‫ڈ‬஼ ሺ࢞ǡ ߲ ఈ ࢞ǡ ߲ ఈ ࢞ǡ Ǥ Ǥ Ǥ ǡ ߲ ఈ ࢞ሻ ൌ ‫࢞ۃܥ‬ǡ ߲ۧ భ

మ

೙

೙

To prove algebraic independence of the system ࢞ǡ ߲ ఈ ࢞ǡ ߲ ఈ ࢞ǡ Ǥ Ǥ Ǥ ǡ ߲ ఈ ࢞ over ‫࢞ۃܥ‬ǡ ߲ۧீ௅ሺ௡ǡ஼ሻ‫ڈ‬஼ it is enough to భ మ ೙ ೙ show algebraic independence of the system ߲ ఈ ࢞ǡ ߲ ఈ ࢞ǡ Ǥ Ǥ Ǥ ǡ ߲ ఈ ࢞ over ‫࢞ۃܥ‬ǡ ߲ۧீ௅ሺ௡ǡ஼ሻ‫ڈ‬஼ . ೙ Let ܲሾࢠଵ ǡ ࢠଶ ǡ Ǥ Ǥ Ǥ ǡ ࢠ௡ ሿ, where ࢠ௜ ൌ ሺ‫ݖ‬ଵ௜ ǡ ‫ݖ‬ଶ௜ ǡ Ǥ Ǥ Ǥ ǡ ‫ݖ‬௡௜ ሻ, be a nonzero polynomial over ‫࢞ۃܥ‬ǡ ߲ۧீ௅ሺ௡ǡ஼ሻ‫ڈ‬஼ such that భ మ ೙ ܲሾ߲ ఈ ࢞ǡ ߲ ఈ ࢞ǡ ǡ Ǥ Ǥ Ǥ ǡ ߲ ఈ ࢞ሿ ൌ Ͳ. Assume, for example, at least one of ‫ݖ‬௡௜ , where ݅ ൌ ͳǡ ݊, occurs in ܲሾࢠଵ ǡ ࢠଶ ǡ Ǥ Ǥ Ǥ ǡ ࢠ௡ ሿ and


ܲሾࢠଵ ǡ ࢠଶ ǡ Ǥ Ǥ Ǥ ǡ ࢠ௡ ሿ ൌ ෍ሺ ‫ݖ‬௡ଵ ሻఉభ ሺ‫ݖ‬௡ଶ ሻఉమ Ǥ Ǥ Ǥ ሺ‫ݖ‬௡௡ ሻఉ೙ ܲఉ ሾࢠଵ ǡ ࢠଶ ǡ Ǥ Ǥ Ǥ ǡ ࢠ௡ ሿ ఉ ೙

௜ ሻ, ݅ ൌ ͳǡ ݊. , where ܲఉ ሾࢠଵ ǡ ࢠଶ ǡ Ǥ Ǥ Ǥ ǡ ࢠ௡ ሿ are polynomials over ‫࢞ۃܥ‬ǡ ߲ۧீ௅ሺ௡ǡ஼ሻ‫ڈ‬஼ in ࢠ௜ ൌ ሺ‫ݖ‬ଵ௜ ǡ ‫ݖ‬ଶ௜ ǡ Ǥ Ǥ Ǥ ǡ ‫ݖ‬௡ିଵ Consider ͳ Ͳ Ͳ ‫ܿ ڮ‬ଵ ‫ې ܿ ڮ Ͳ ͳ Ͳۍ‬ ଶ ‫ێ‬ ‫ۑ‬ ݄ ൌ ‫ܿ ڮ ͳ Ͳ Ͳێ‬ଷ ‫ܮܩ א ۑ‬ሺ݊ǡ ‫ܥ‬ሻ ‫ۑڭ‬ ‫ڭ ڭ ڭ ڭێ‬ ‫ܿ ڮ Ͳ Ͳ Ͳۏ‬௡ ‫ے‬

For such ݄ one has ݄࢞ ൌ ‫ݔ‬. So far as the coefficients of ܲሾࢠଵ ǡ ࢠଶ ǡ Ǥ Ǥ Ǥ ǡ ࢠ௡ ሿ are ‫ܮܩ‬ሺ݊ǡ ‫ܥ‬ሻ ‫ ܥ ڈ‬௡ - invariant, భ మ ೙ substitution ݄࢞ for ࢞ into ܲሾ߲ ఈ ࢞ǡ ߲ ఈ ࢞ǡ ǡ Ǥ Ǥ Ǥ ǡ ߲ ఈ ࢞ሿ ൌ Ͳ implies that ௡

௡ భ

௡ మ

೙

భ

మ

೙

෍ሺ ෍ ܿ௜ ߲ ఈ ‫ݔ‬௜ ሻఉభ ሺ෍ ܿ௜ ߲ ఈ ‫ݔ‬௜ ሻఉమ Ǥ Ǥ Ǥ ሺ෍ ܿ௜ ߲ ఈ ‫ݔ‬௜ ሻఉ೙ ܲఉ ሾ߲ ఈ ࢞ǡ ߲ ఈ ࢞ǡ Ǥ Ǥ Ǥ ǡ ߲ ఈ ࢞ሿ ൌ ͲǤ ఉ

௜ୀଵ

௜ୀଵ

௜ୀଵ

Therefore due to zero characteristic of ‫ ܨ‬for variables ‫ݕ‬ଵ ǡ ‫ݕ‬ଶ ǡ Ǥ Ǥ Ǥ ǡ ‫ݕ‬௡ one has ௡

௡ ఈభ

௡ ఈమ

೙

௜ୀଵ

௜ୀଵ

ఉమ

భ

మ

೙

෍ሺ ෍ ‫ݕ‬௜ ߲ ‫ݔ‬௜ ሻ ሺ෍ ‫ݕ‬௜ ߲ ‫ݔ‬௜ ሻ Ǥ Ǥ Ǥ ሺ෍ ‫ݕ‬௜ ߲ ఈ ‫ݔ‬௜ ሻఉ೙ ܲఉ ሾ߲ ఈ ࢞ǡ ߲ ఈ ࢞ǡ Ǥ Ǥ Ǥ ǡ ߲ ఈ ࢞ሿ ൌ Ͳ ఉ

ఉభ

(1)

௜ୀଵ

Now consider the ring ‫࢞ۃܥ‬ǡ ߲ۧሾ‫ݕ‬ଵ ǡ ‫ݕ‬ଶ ǡ Ǥ Ǥ Ǥ ǡ ‫ݕ‬௡ ሿ with respect to the differential operators డ డ డ ǡ ߲ଶ ൌ ǡ Ǥ Ǥ Ǥ ǡ ߲௡ ൌ . It is clear that its constant ring is ‫࢞ۃܥ‬ǡ ߲ۧ i.e. ߲ଵ ൌ డ௬భ

డ௬మ

డ௬೙

‫࢞ۃܥ‬ǡ ߲ۧ ൌ ሼܽ ‫࢞ۃܥ א‬ǡ ߲ۧሾ‫ݕ‬ଵ ǡ ‫ݕ‬ଶ ǡ Ǥ Ǥ Ǥ ǡ ‫ݕ‬௡ ሿǣ ߲ଵ ܽ ൌ ߲ଶ ܽ ൌǤ Ǥ Ǥ ൌ ߲௡ ܽ ൌ ͲሽǤ ௡ Introduce new differential operators ߜ௜ ൌ σ௝ୀଵ ݂௜௝డ ‫߲ۧ࢞ۃ‬௝ , where ݅ ൌ ͳǡ ݊, భ

మ

೙

ሺ݂௜௝డ ‫ۧ࢞ۃ‬ሻ௜ǡ௝ୀଵǡ௡ ൌ ሾ߲ ఈ ࢞ǡ ߲ ఈ ࢞ǡ Ǥ Ǥ Ǥ ǡ ߲ ఈ ࢞ሿିଵ The following are evident: a) The constant ring of ሺ‫࢞ۃܥ‬ǡ ߲ۧሾ‫ݕ‬ଵ ǡ ‫ݕ‬ଶ ǡ Ǥ Ǥ Ǥ ǡ ‫ݕ‬௡ ሿǡ ߜሻ, where ߜ ൌ ሺߜଵ ǡ ߜଶ ǡ Ǥ Ǥ Ǥ ǡ ߜ௡ ሻ is the same ‫࢞ۃܥ‬ǡ ߲ۧ, ೖ b) ߜ௝ ሺσ௡௜ୀଵ ‫ݕ‬௜ ߲ ఈ ‫ݔ‬௜ ሻ is equal to Ͳ whenever ݆ ് ݇ and it is equal to 1 if ݆ ൌ ݇, where ݆ǡ ݇ ൌ ͳǡ ݊ . Now if one assumes that ߚ ଴ ൌ ሺߚଵ଴ ǡ Ǥ Ǥ Ǥ ǡ ߚ௡଴ ሻ is a such one for which భ

మ

೙

ȁߚ ଴ ȁ ൌ ᩸ሼȁߚȁǣ ܲఉ ሾ߲ ఈ ࢞ǡ ߲ ఈ ࢞ǡ Ǥ Ǥ Ǥ ǡ ߲ ఈ ࢞ሿ ് Ͳሽ ఉబ ఉబ

ఉబ

భ

మ

೙

and applies ߜଵ భ ߜଶ మ Ǥ Ǥ Ǥ ߜ௡ ೙ to equality (1) he comes to a contradiction ܲఉబ ሾ߲ ఈ ࢞ǡ ߲ ఈ ࢞ǡ Ǥ Ǥ Ǥ ǡ ߲ ఈ ࢞ሿ ൌ Ͳ. Remark 2.1: If one is interested only in differential rational invariants of subgroups of the general liner group ‫ܮܩ‬ሺ݊ǡ ‫ܥ‬ሻ then the following analogue of this result can be used: భ

మ

೙షభ

Theorem 2.1’: The equality ‫࢞ۃܥ‬ǡ ߲ۧீ௅ሺ௡ǡ஼ሻ ሺ࢞ǡ ߲ ఈ ࢞ǡ ߲ ఈ ࢞ǡ Ǥ Ǥ Ǥ ǡ ߲ ఈ ࢞ሻ ൌ ‫࢞ۃܥ‬ǡ ߲ۧ is valid and moreover the భ మ ೙షభ system consisting of components of ࢞ǡ ߲ ఈ ࢞ǡ ߲ ఈ ࢞ǡ Ǥ Ǥ Ǥ ǡ ߲ ఈ ࢞ is algebraic independent over ‫࢞ۃܥ‬ǡ ߲ۧீ௅ሺ௡ǡ஼ሻ . A proof of this result can be done in a similar way as in Theorem 2.1. So due to this Theorem ೙ భ ೙ ‫࢞ۃܥ‬ǡ ߲ۧு ൌ ‫ݔۃܥ‬ǡ ߲ۧீ௅ሺ௡ǡ஼ሻ‫ڈ‬஼ ሺ࢞ǡ ߲ ఈ ࢞ǡ Ǥ Ǥ Ǥ ǡ ߲ ఈ ࢞ሻு భ

೙

೙

and the system ࢞ǡ ߲ ఈ ࢞ǡ Ǥ Ǥ Ǥ ǡ ߲ ఈ ࢞ is algebraic independent over ‫࢞ۃܥ‬ǡ ߲ۧீ௅ሺ௡ǡ஼ሻ‫ڈ‬஼ . Note that every element of the ೙ field ‫࢞ۃܥ‬ǡ ߲ۧீ௅ሺ௡ǡ஼ሻ‫ڈ‬஼ is a fixed element for the group ‫ܪ‬. Therefore if one wants to have a system of ߲differential generators of ‫࢞ۃܥ‬ǡ ߲ۧு over ‫ ܥ‬he can do the following: ೙ 1. Find any system of generators (over C) of the ߲-differential field ‫࢞ۃܥ‬ǡ ߲ۧீ௅ሺ௡ǡ஼ሻ‫ڈ‬஼ For example, one can take for it the system presented in Theorem 2.2 generators 2. Find any system of ordinary algebraic generators of the field


೙

‫࢞ۃܥ‬ǡ ߲ۧீ௅ሺ௡ǡ஼ሻ‫ڈ‬஼ ሺࢠଵ ǡ ࢠଶ ǡ Ǥ Ǥ Ǥ ǡ ࢠ௡ାଵ ሻு , where ࢠ௜ ൌ ሺ‫ݖ‬ଵ௜ ǡ ‫ݖ‬ଶ௜ ǡ Ǥ Ǥ Ǥ ǡ ‫ݖ‬௡௜ ሻ , ݅ ൌ ͳǡ ݊ ൅ ͳ, and the action of ‫ ܪ‬is defined as: ሺሺ݄ǡ ݄଴ ሻǡ ሺࢠଵ ǡ ࢠଶ ǡ Ǥ Ǥ Ǥ ǡ ࢠ௡ାଵ ሻሻ ՜ ሺࢠଵ ݄ ൅ ݄଴ ǡ ࢠଶ ݄ǡ Ǥ Ǥ Ǥ ǡ ࢠ௡ାଵ ݄ሻ For example, let it be ߮ଵ ሺࢠଵ ǡ ࢠଶ ǡ Ǥ Ǥ Ǥ ǡ ࢠ௡ାଵ ሻǡ ߮ଶ ሺࢠଵ ǡ ࢠଶ ǡ Ǥ Ǥ Ǥ ǡ ࢠ௡ାଵ ሻǡ Ǥ Ǥ Ǥ ǡ ߮௞ ሺࢠଵ ǡ ࢠଶ ǡ Ǥ Ǥ Ǥ ǡ ࢠ௡ାଵ ሻ. Then the union of the ೙ system of generators of ሺ‫࢞ۃܥ‬ǡ ߲ۧீ௅ሺ௡ǡ஼ሻ‫ڈ‬஼ Ǣ ߲ሻ with భ

೙

భ

೙

భ

೙

ሼ߮ଵ ሺ࢞ǡ ߲ ఈ ࢞ǡ Ǥ Ǥ Ǥ ǡ ߲ ఈ ࢞ሻǡ ߮ଶ ሺ࢞ǡ ߲ ఈ ࢞ǡ Ǥ Ǥ Ǥ ǡ ߲ ఈ ࢞ሻǡ Ǥ Ǥ Ǥ ǡ ߮௞ ሺ࢞ǡ ߲ ఈ ࢞ǡ Ǥ Ǥ Ǥ ǡ ߲ ఈ ࢞ሻሽ can be taken as a system of generators of the ߲-differential field ‫࢞ۃܥ‬ǡ ߲ۧு over ‫ܥ‬. Remark 2.2: In the case of ݉ ൌ ͳ to find a system of generators of the field ೙

‫࢞ۃܥ‬ǡ ߲ۧீ௅ሺ௡ǡ஼ሻ‫ڈ‬஼ ሺࢠଵ ǡ ࢠଶ ǡ Ǥ Ǥ Ǥ ǡ ࢠ௡ାଵ ሻு it is enough to find generators of ‫ܥ‬ሺࢠଵ ǡ ࢠଶ ǡ Ǥ Ǥ Ǥ ǡ ࢠ௡ାଵ ሻு due to the fact that in this case the ߲- differential field ೙ ‫࢞ۃܥ‬ǡ ߲ۧீ௅ሺ௡ǡ஼ሻ‫ڈ‬஼ admits ߲-algebraic independent system of generators over ‫[ ܥ‬4]. It seems true that in ݉ ൐ ͳ ೙ case ‫࢞ۃܥ‬ǡ ߲ۧீ௅ሺ௡ǡ஼ሻ‫ڈ‬஼ doesn’t admit such system of generators. Therefore the approach used in [4] doesn’t work ೙ here. Under additional assumption we show that to find generators of ‫࢞ۃܥ‬ǡ ߲ۧீ௅ሺ௡ǡ஼ሻ‫ڈ‬஼ still it is enough to find generators of ‫ܥ‬ሺࢠଵ ǡ ࢠଶ ǡ Ǥ Ǥ Ǥ ǡ ࢠ௡ାଵ ሻு . Proposition 2.1: Let ‫ ܥ‬be an algebraic closed field, ‫ ܭ ؿ ܥ‬be an extension of fields, ܲሾ࢞ሿ ‫ܭ א‬ሾ࢞ሿ be a nonzero polynomial and ݄ ‫ܮܩ א‬ሺ݊ǡ ‫ܥ‬ሻ. If ܲሾ݄࢞ሿ ൌ ݇ܲሾ࢞ሿ, where ݇ ‫ܭ א‬, then ݇ has to be in ‫ܥ‬. Proof: Consider finite dimensional space ஼ ሺሺ‫݌‬௜ ሻ௜ ǡ ሺ݇‫݌‬௜ ሻ௜ ሻ, where ሺ‫݌‬௜ ሻ௜ stands for set of all coefficients of ܲሾ࢞ሿ, any linear basis ሺ݇௝ ሻ௝ of it over ‫ܥ‬, ‫ܥ‬-vector space ܸ ൌ ஼ ሺሺ݇௝ ࢞ఈ ሻ௝ǡȁఈȁஸௗ௘௚௉ሾ࢞ሿ ሻ and ‫ܥ‬-linear map ‫ܮ‬ǣ ܸ ՜ ܸ for which ‫ܮ‬ǣ ݇௝ ࢞ఈ հ ݇௝ ሺ݄࢞ሻఈ . In this case equality ܲሾ݄࢞ሿ ൌ ݇ܲሾ࢞ሿ means that ݇ is an eigenvalue for ‫ܮ‬ and therefore ݇ ‫ ܥ א‬as far as ‫ ܥ‬is algebraic closed. Proposition 2.2: Let ‫ ܭ ؿ ܥ‬be an extension of fields, ‫ ܪ‬be a subgroup of ‫ܮܩ‬ሺ݊ǡ ‫ܥ‬ሻ. If for any character ߯ǣ ‫ ܪ‬՜ ‫ ܭ‬its range is in ‫ ܥ‬or ‫ ܥ‬is an algebraic closed field then ‫ܭ‬ሺ࢞ሻு ൌ ‫ܭ‬ሺ‫ܥ‬ሺ࢞ሻு ሻ Proof: Let

௉ሾ࢞ሿ ொሾ࢞ሿ

‫ܭ א‬ሺ࢞ሻு be an irreducible ratio of polynomials. Due to equality

௉ሾ࢞௛ሿ ொሾ࢞௛ሿ

ൌ


one can

conclude that ܲሾ݄࢞ሿ ൌ ߯ሺ݄ሻܲሾ࢞ሿ, ܳሾ݄࢞ሿ ൌ ߯ሺ݄ሻܳሾ࢞ሿ, where ݄ ‫ܪ א‬, and ߯ǣ ‫ ܪ‬՜ ‫ ܭ‬is a commutative character of ‫ܪ‬. It implies that for each homogenous component ܲ௜ ሾ࢞ሿ (ܳ௜ ሾ࢞ሿ) of ܲሾ࢞ሿ (respect. ܳሾ࢞ሿ) the equality ܲ௜ ሾ݄࢞ሿ ൌ ߯ሺ݄ሻܲ௜ ሾ࢞ሿ (respect. ܳ௜ ሾ݄࢞ሿ ൌ ߯ሺ݄ሻܳ௜ ሾ࢞ሿ) is valid as well. According to the assumption of Proposition 2.2 and Proposition 2.1 the range of ߯ is in ‫ܥ‬. Consider expansion ܲ௜ ሾ࢞ሿ ൌ σ ݇௝ ܲᇱ ௜ǡ௝ ሾ࢞ሿ (ܳ௜ ሾ࢞ሿ ൌ σ ݇௝ ܳᇱ ௜ǡ௝ ሾ࢞ሿ), where ሺ݇௝ ሻ௝ is a ‫ܥ‬-linear independent system of elements of ‫ܭ‬, ܲᇱ ௜ǡ௝ ሾ࢞ሿ ( respect. ܳᇱ ௜ǡ௝ ሾ࢞ሿ) are homogenous polynomials over ‫ܥ‬. Now one can conclude that for all ሺ݅ǡ ݆ሻ the equalities ܲᇱ ௜ǡ௝ ሾ݄࢞ሿ ൌ ߯ሺ݄ሻܲᇱ ௜ǡ௝ ሾ࢞ሿ ( respect. ܳᇱ ௜ǡ௝ ሾ݄࢞ሿ ൌ ߯ሺ݄ሻܳᇱ ௜ǡ௝ ሾ࢞ሿ) are also true. Fixing any nonzero ܳᇱ ௜బ ǡ௝బ ሾ‫ݔ‬ሿ and representation ܲሾ࢞ሿ ൌ ܳሾ࢞ሿ

indicate that


‫ܭ א‬ሺ‫ܥ‬ሺ࢞ሻு ሻ because of

௉ᇲ ೔ǡೕሾ࢞ሿ ொᇲ

೔బ ǡೕబ ሾ࢞ሿ

σ ݇௝ σ ݇௝

ܲᇱ ௜ǡ௝ ሾ࢞ሿ ܳᇱ ௜బ ǡ௝బ ሾ࢞ሿ ܳᇱ ௜ǡ௝ ሾ࢞ሿ ܳᇱ ௜బ ǡ௝బ ሾ࢞ሿ

‫ܥ א‬ሺ࢞ሻு ,

௉ᇲ ೔ǡೕ ሾ࢞ሿ ொᇲ ೔ ǡೕ ሾ࢞ሿ బ బ

‫ܥ א‬ሺ࢞ሻு .

Theorem 2.1 with Proposition 2.2 show that at least in algebraic closed constant field case for any subgroup ‫ ܪ‬of the Affine group a finite system of generators of the ߲-differential field ‫࢞ۃܥ‬ǡ ߲ۧு can be constructed by the use of its algebraic (not differential) invariants over ‫ܥ‬. The same is true for ‫ ܥ‬ൌ ܴ, ‫ ܪ‬ൌ ܱሺ݊ǡ ܴሻ and ‫ ܪ‬ൌ


ܱܵሺ݊ǡ ܴሻ cases due to the fact that the group ܱሺ݊ǡ ܴሻ has only one nontrivial commutative character ߯ሺ݄ሻ ൌ ሺ ݄ሻ ൌ േͳ ‫ ܴ א‬and ܱܵሺ݊ǡ ܴሻ has no nontrivial commutative character. Now as an application of Theorem 2.1 let us describe ߲-differential field ‫࢞ۃܥ‬ǡ ߲ۧு for some classic subgroups ‫ ܪ‬of the affine group by providing a finite system of generators. In future let the matrix ܺ ൌ ܺሺ࢞ሻ ೔ stand for ܺሺ࢞ሻ ൌ ሺ߲ ఈ ࢞ሻ௜ୀଵǡǤǤǤǡ௡ , where ߙ ଵ ൌ ሺͳǡͲǡ Ǥ Ǥ Ǥ ǡͲሻǡ ߙ ଶ ൌ ሺͲǡͳǡ Ǥ Ǥ Ǥ ǡͲሻǡ Ǥ Ǥ Ǥ ǡ ߙ ௠ ൌ ሺͲǡͲǡ Ǥ Ǥ Ǥ ǡͳሻǡ ߙ ௠ାଵ ǡ Ǥ Ǥ Ǥ Ǥ ǡ ߙ ௡ are any different nonzero ݉-tuples with nonnegative integer entries. If ݊ ൏ ݉ it is assumed that ܺሺ࢞ሻ consists ೙ of the first ݊ rows of it. Note that all components of ߲௜ ܺሺ࢞ሻܺሺ࢞ሻିଵ are in ‫࢞ۃܥ‬ǡ ߲ۧீ௅ሺ௡ǡ஼ሻ‫ڈ‬஼ at any ݅ ൌ ͳǡʹǡ Ǥ Ǥ Ǥ ǡ ݉. Theorem 2.2: a) If ݊ ൐ ݉ (݊ ൑ ݉) then the system of entries of matrices ሺ߲௜ ܺܺ ିଵ ሻଵஸ௜ஸ௠ ሺǤ ሺ߲௜ ܺܺ ିଵ ሻଵஸ௜ஸ௠ ǡ ሺ߲௜ ࢞ܺ ିଵ ሻ௡ାଵஸ௜ஸ௠ ሻ ೙

generates ߲-differential field C‫࢞ۃ‬ǡ ߲ۧீ௅ሺ௡ǡ஼ሻ‫ڈ‬஼ over ‫ܥ‬. b) If ݊ ൐ ݉ (݊ ൑ ݉) then the system of entries of matrices ሺ߲௜ ܺܺ ିଵ ሻ௜ୀଵǡଶǡǤǤǤǡ௠ and ܺ ሺǤ ሺ߲௜ ܺܺ ିଵ ሻ௜ୀଵǡଶǡǤǤǤǡ௠ ǡ ሺ߲௜ ࢞ܺ ିଵ ሻ௜ୀ௡ାଵǡǤǤǤǡ௠ ǡ ܺሻ ೙

generates ߲-differential field ‫࢞ۃܥ‬ǡ ߲ۧௌ௅ሺ௡ǡ஼ሻ‫ڈ‬஼ over ‫ܥ‬. Proof: Note that if the equality ݂‫ ۧ࢞ۃ‬ൌ ݂‫ ࢞ۃ‬൅ ݄଴ ۧ is true for any ݄଴ ‫ ܥ א‬௡ then variable ࢞ does not occur in ݂ and therefore ݂‫ ۧ࢞ۃ‬ൌ ݃ሺሺ߲ ఈ ࢞ሻȁఈȁவ଴ ሻ. a)

೙

If ݂‫࢞ۃܥ א ۧ࢞ۃ‬ǡ ߲ۧீ௅ሺ௡ǡ஼ሻ‫ڈ‬஼ then ݂‫ ۧ࢞ۃ‬ൌ ݃ሺሺ߲ ఈ ࢞ሻȁఈȁவ଴ ሻ and ݃ሺሺ߲ ఈ ࢞ሻȁఈȁவ଴ ሻ ൌ ݃ሺሺ߲ ఈ ሺ݄࢞ ൅ ݄଴ ሻሻȁఈȁவ଴ ሻ ൌ ݃ሺሺ߲ ఈ ݄࢞ሻȁఈȁவ଴ ሻ ൌ ݃ሺሺሺ߲ ఈ ࢞ሻ݄ሻȁఈȁவ଴ ሻ

As far as the characteristic of ‫ ܨ‬is zero one has ݃ሺሺ߲ ఈ ࢞ሻȁఈȁவ଴ ሻ ൌ ݃ሺሺሺ߲ ఈ ࢞ሻܶሻȁఈȁவ଴ ሻ for any ܶ ൌ ሺ‫ݐ‬௜௝ ሻ௜ǡ௝ୀଵǡǤǤǤǡ௡ . So one can substitute for ܶ the matrix ܺ ିଵ ሺ࢞ሻ and by induction, due to the identity ሺ߲௜ ߲ ఈ ࢞ሻܺ ିଵ ൌ ߲௜ ሺ߲ ఈ ࢞ܺ ିଵ ሻ ൅ ሺ߲ ఈ ࢞ܺ ିଵ ሻ߲௜ ܺܺ ିଵ , conclude that indeed the system of components of the matrices ሺ߲௜ ࢞ܺ ିଵ ሻଵஸ௜ஸ௠ ǡ ሺ߲௜ ܺܺ ିଵ ሻଵஸ௜ஸ௠ ೙

generates ‫࢞ۃܥ‬ǡ ߲ۧீ௅ሺ௡ǡ஼ሻ‫ڈ‬஼ over ‫ ܥ‬as a ߲-differential field. If ݊ ൐ ݉ then ߲௜ ࢞ܺ ିଵ ൌ ࢋ௜ for all ݅ ൌ ͳǡʹǡ Ǥ Ǥ Ǥ ǡ ݉ and if ݊ ൑ ݉ then ߲௜ ࢞ܺ ିଵ ൌ ࢋ௜ for all ݅ ൌ ͳǡʹǡ Ǥ Ǥ Ǥ ǡ ݊, where ࢋ௜ stands for the ݅-th row of the ݊-order identity ೙ matrix. Therefore in first case the ߲-differential field ‫࢞ۃܥ‬ǡ ߲ۧீ௅ሺ௡ǡ஼ሻ‫ڈ‬஼ is generated by the system of ିଵ components of ሺ߲௜ ܺܺ ሻ௜ୀଵǡǤǤǤǡ௠ and in the second case it is generated by the system of components of ሺ߲௜ ࢞ܺ ିଵ ሻ௜ୀ௡ାଵǡǤǤǤǡ௠ ǡ ሺ߲௜ ܺܺ ିଵ ሻ௜ୀଵǡǤǤǤǡ௠ over ‫ܥ‬. b) Proof can be done similar to part a) way. Substitute for the ܶ the matrix ܺ ିଵ ሺ࢞ሻ‫ܦ‬ଵ ሺሺ ܺሻሻ and by induction , due to the identity ሺ߲௜ ߲ ఈ ࢞ሻܺ ିଵ ‫ܦ‬ଵ ሺሺ ܺሻሻ ൌ ߲௜ ሺ߲ ఈ ࢞ܺ ିଵ ሻ‫ܦ‬ଵ ሺሺ ܺሻሻ ൅ ሺ߲ ఈ ࢞ܺ ିଵ ሻ߲௜ ܺܺ ିଵ ‫ܦ‬ଵ ሺሺ ܺሻሻ , conclude that indeed the system of components of the matrices ሺ߲௜ ࢞ܺ ିଵ ሻଵஸ௜ஸ௡ ǡ ሺ߲௜ ܺܺ ିଵ ሻଵஸ௜ஸ௠ ሺ ܺሻ ೙

generates ‫࢞ۃܥ‬ǡ ߲ۧௌ௅ሺ௡ǡ஼ሻ‫ڈ‬஼ over ‫ ܥ‬as a ߲-differential field, where ‫ܦ‬ଵ ሺሺ ܺሻሻ ൌ ‫݃ܽ݅ܦ‬ሺ ܺ ǡ ͳǡͳǡ Ǥ Ǥ Ǥ ǡͳሻ. So ೙ for ‫࢞ۃܥ‬ǡ ߲ۧௌ௅ሺ௡ǡ஼ሻ‫ן‬஼ case one should join ܺ ሺ࢞ሻ to the system of generators of part a).


In ‫ ܥ‬ൌ ܴ, ݊ ൌ ݉ case in [8] it is shown that the system of components of ሺ߲௜ ܺܺ ିଵ ሻଵஸ௜ஸ௠ -the Christoffel symbols of the second kind generates every Affine-invariant function, in particular, every Affine-invariant differential rational function is a function of these invariants as far as there "generates" means as a function in common sense. In this case we are stating more stronger thing: Every Affine-invariant differential rational function is a differential rational function of these invariants. The same can be said with respect to ೙ ‫࢞ۃܥ‬ǡ ߲ۧௌ௅ሺ௡ǡோሻ‫ڈ‬ோ as well. For the next theorem ሺ‫כ‬ǡ‫כ‬ሻ stands for the dot product and it is assumed that ‫ ܥ‬is any algebraic closed field or ‫ ܥ‬ൌ ܴ. Theorem 2.3: a) If ݊ ൐ ݉ (݊ ൑ ݉) then the system ೔

ೕ

೔

ೕ

ሺ߲ ఈ ࢞ǡ ߲ ఈ ࢞ሻଵஸ௜ஸ௝ஸ௡ ǡ ሺ߲௞ ߲ ఈ ࢞ǡ ߲ ఈ ࢞ሻଵஸ௜ஸ௝ஸ௡ǡଵஸ௞ஸ௠ ሺǤ ሺ߲௜ ࢞ǡ ߲௝ ࢞ሻଵஸ௜ஸ௝ஸ௡ ǡ ሺ߲௜ ࢞ǡ ߲௝ ࢞ሻ௡ାଵஸ௜ஸ௠ǡଵஸ௝ஸ௡ ሻ ೙

generates ߲-differential field ‫࢞ۃܥ‬ǡ ߲ۧைሺ௡ǡ஼ሻ‫ڈ‬஼ over ‫ܥ‬. b) If ݊ ൐ ݉ (݊ ൑ ݉) then the system ೔

ೕ

೔

ೕ

ሺ߲ ఈ ࢞ǡ ߲ ఈ ࢞ሻଵஸ௜ஸ௝ஸ௡ǡ௜ା௝ழଶ௡ ǡ ሺ߲௞ ߲ ఈ ࢞ǡ ߲ ఈ ࢞ሻଵஸ௜ஸ௝ஸ௡ǡଵஸ௞ஸ௠ ǡ ܺ ሺǤ ሺ߲௜ ࢞ǡ ߲௝ ࢞ሻଵஸ௜ஸ௝ஸ௡ǡ௜ା௝ழଶ௡ ǡ ሺ߲௜ ࢞ǡ ߲௝ ࢞ሻ௡ାଵஸ௜ஸ௠ǡଵஸ௝ஸ௡ ǡ ܺሻ ೙

generates ߲-differential field ‫ݔۃܥ‬ǡ ߲ۧௌைሺ௡ǡ஼ሻ‫ڈ‬஼ over ‫ܥ‬. Proof: To prove a) note that if ሺ‫ݖ‬௝௜ ሻଵஸ௜ஸ௡ǡଵஸ௝ஸ௡ is a system of algebraic independent variables over ‫ ܥ‬then ‫ܥ‬ሺܼሻைሺ௡ǡ஼ሻ ൌ ‫ܥ‬ሺሺ‫ ݖ‬௜ ሻଵஸ௜ஸ௡ ሻைሺ௡ǡ஼ሻ ൌ ‫ܥ‬ሺሺ‫ ݖ‬௜ ǡ ‫ ݖ‬௝ ሻଵஸ௜ஸ௝ஸ௡ ሻ, where ܼ is matrix consisting of rows ࢠଵ ǡ ࢠଶ ǡ Ǥ Ǥ Ǥ ǡ ࢠ௡ , ࢠ௜ ൌ ሺ‫ݖ‬ଵ௜ ǡ ‫ݖ‬ଶ௜ ǡ Ǥ Ǥ Ǥ ǡ ‫ݖ‬௡௜ ሻ. Indeed if the orthogonal system of vectors ‫ܨ‬ଵ ሺܼሻǡ ‫ܨ‬ଶ ሺܼሻǡ Ǥ Ǥ Ǥ ǡ ‫ܨ‬௡ ሺܼሻ is the result of GramSchmidt process over the system of column vectors ሺࢠଵ ሻ௧ ǡ ሺࢠଶ ሻ௧ ǡ Ǥ Ǥ Ǥ ǡ ሺࢠ௡ ሻ௧ then ሺ‫ܨ‬ଵ ሺܼሻǡ ‫ܨ‬ଶ ሺܼሻǡ Ǥ Ǥ Ǥ ǡ ‫ܨ‬௡ ሺܼሻሻ ൌ ܼ ௧ ܶሺܼሻ , where entries of ܶሺܼሻሻ are rational functions in ሺࢠ௜ ǡ ࢠ௝ ሻଵஸ௜ஸ௝ஸ௡ over integers. Moreover the system ሺͳǡ ሺටሺ‫ܨ‬௜భ ሺܼሻǡ ‫ܨ‬௜భ ሺܼሻሻටሺ‫ܨ‬௜మ ሺܼሻǡ ‫ܨ‬௜మ ሺܼሻሻǤ Ǥ Ǥ ටሺ‫ܨ‬௜ೖ ሺܼሻǡ ‫ܨ‬௜ೖ ሺܼሻሻሻଵஸ௜భழ௜మ ழǤǤǤ௜ೖ ஸ௡ ሻ is a basis for the vector space ‫ܥ‬ሺܼሻሺඥሺ‫ܨ‬ଵ ሺܼሻǡ ‫ܨ‬ଵ ሺܼሻሻǡ ඥሺ‫ܨ‬ଶ ሺܼሻǡ ‫ܨ‬ଶ ሺܼሻሻǡ Ǥ Ǥ Ǥ ǡ ඥሺ‫ܨ‬௡ ሺܼሻǡ ‫ܨ‬௡ ሺܼሻሻሻ over ‫ܥ‬ሺܼሻ. Consider orthogonal matrix ሺ‫ܨ‬ଵ ሺܼሻǡ ‫ܨ‬ଶ ሺܼሻǡ Ǥ Ǥ Ǥ ǡ ‫ܨ‬௡ ሺܼሻሻ‫ି ܦ‬ଵ ሺܼሻ ൌ ܼ ௧ ܶሺܼሻ‫ି ܦ‬ଵ ሺܼሻ , where ‫ܦ‬ሺܼሻ ൌ ‫݃ܽ݅ܦ‬ሺඥሺ‫ܨ‬ଵ ሺܼሻǡ ‫ܨ‬ଵ ሺܼሻሻǡ ඥሺ‫ܨ‬ଶ ሺܼሻǡ ‫ܨ‬ଶ ሺܼሻሻǡ Ǥ Ǥ Ǥ ǡ ඥሺ‫ܨ‬௡ ሺܼሻǡ ‫ܨ‬௡ ሺܼሻሻሻ. If ݂ሺܼሻ ‫ܥ א‬ሺܼሻைሺ௡ǡ஼ሻ then ݂ሺܼሻ ൌ ݂ሺܼܼ ௧ ܶሺܼሻ‫ି ܦ‬ଵ ሺܼሻሻ that is ݂ሺܼሻ ‫ܥ א‬ሺሺࢠ௜ ǡ ࢠ௝ ሻଵஸ௜ஸ௝ஸ௡ ሻሺඥሺ‫ܨ‬ଵ ሺܼሻǡ ‫ܨ‬ଵ ሺܼሻሻǡ ඥሺ‫ܨ‬ଶ ሺܼሻǡ ‫ܨ‬ଶ ሺܼሻሻǡ Ǥ Ǥ Ǥ ǡ ඥሺ‫ܨ‬௡ ሺܼሻǡ ‫ܨ‬௡ ሺܼሻሻሻ which implies that ݂ሺܼሻ ‫ܥ א‬ሺሺࢠ௜ ǡ ࢠ௝ ሻଵஸ௜ஸ௝ஸ௡ ሻ. Now due to Theorem 2.1, Theorem 2.2, the Propositions and this fact for ݊ ൐ ݉ case one has ೙

೙

భ

మ

೙

೙

‫࢞ۃܥ‬ǡ ߲ۧைሺ௡ǡ஼ሻ‫ڈ‬஼ ൌ ‫࢞ۃܥ‬ǡ ߲ۧீ௅ሺ௡ǡ஼ሻ‫ڈ‬஼ ሺ࢞ǡ ߲ ఈ ࢞ǡ ߲ ఈ ࢞ǡ Ǥ Ǥ Ǥ ǡ ߲ ఈ ࢞ሻைሺ௡ǡ஼ሻ‫ڈ‬஼ ൌ ೙

భ

మ

೙

೙

೔

ೕ

‫࢞ۃܥ‬ǡ ߲ۧீ௅ሺ௡ǡ஼ሻ‫ڈ‬஼ ሺ‫ܥ‬ሺ߲ ఈ ࢞ǡ ߲ ఈ ࢞ǡ Ǥ Ǥ Ǥ ǡ ߲ ఈ ࢞ሻைሺ௡ǡ஼ሻ ሻ ൌ ‫࢞ۃܥ‬ǡ ߲ۧீ௅ሺ௡ǡ஼ሻ‫ڈ‬஼ ሺሺ߲ ఈ ࢞ǡ ߲ ఈ ࢞ሻଵஸ௜ஸ௝ஸ௡ ሻ ൌ


೔

ೕ

‫ۃܥ‬ሺ߲௜ ܺܺ ିଵ ሻଵஸ௜ஸ௠ ǡ ሺ߲ ఈ ࢞ǡ ߲ ఈ ࢞ሻଵஸ௜ஸ௝ஸ௡ ۧ Taking into account the equality ߲௜ ܺܺ ିଵ ൌ ߲௜ ܺܺ ௧ ሺܺܺ ௧ ሻିଵ one can say that the system of components of ೙ matrices ሺ߲௜ ܺܺ ௧ ሻଵஸ௜ஸ௠ ǡ ܺܺ ௧ generates ‫࢞ۃܥ‬ǡ ߲ۧைሺ௡ǡ஼ሻ‫ڈ‬஼ . Therefore in ݊ ൐ ݉ case the system ೔

ೕ

೔

ೕ

ሺ߲ ఈ ࢞ǡ ߲ ఈ ࢞ሻଵஸ௜ஸ௝ஸ௡ ǡ ሺ߲௞ ߲ ఈ ࢞ǡ ߲ ఈ ࢞ሻଵஸ௜ஸ௝ஸ௡ǡଵஸ௞ஸ௠ ೙

generates ߲-differential field ‫࢞ۃܥ‬ǡ ߲ۧைሺ௡ǡ஼ሻ‫ڈ‬஼ over ‫ܥ‬. In the same way for ݊ ൑ ݉ case one has ೙

೙

‫࢞ۃܥ‬ǡ ߲ۧைሺ௡ǡ஼ሻ‫ڈ‬஼ ൌ ‫࢞ۃܥ‬ǡ ߲ۧீ௅ሺ௡ǡ஼ሻ‫ڈ‬஼ ሺሺ߲௜ ࢞ǡ ߲௝ ࢞ሻଵஸ௜ஸ௝ஸ௡ ሻ ൌ ‫ۃܥ‬ሺ߲௜ ܺܺ ିଵ ሻଵஸ௜ஸ௠ ǡ ሺ߲௜ ࢞ܺ ିଵ ሻ௡ାଵஸ௜ஸ௠ ǡ ሺ߲௜ ࢞ǡ ߲௝ ࢞ሻଵஸ௜ஸ௝ஸ௡ ۧ As far as ߲௜ ܺܺ ିଵ ൌ ߲௜ ܺܺ ௧ ሺܺܺ ௧ ሻିଵ , ߲௜ ࢞ܺ ିଵ ൌ ߲௜ ࢞ܺ ௧ ሺܺܺ ௧ ሻିଵ one can say that the system of components of ೙ matrices ሺ߲௜ ܺܺ ௧ ሻଵஸ௜ஸ௠ ǡ ሺ߲௜ ࢞ܺ ௧ ሻ௡ାଵஸ௜ஸ௠ ǡ ܺܺ ௧ generates ‫࢞ۃܥ‬ǡ ߲ۧைሺ௡ǡ஼ሻ‫ڈ‬஼ . Note that the entries ܺܺ ௧ are ሺሺ߲௜ ࢞ǡ ߲௝ ࢞ሻଵஸ௜ஸ௝ஸ௡ ሻ and the identity ͳ ሺ߲௜ ߲௝ ࢞ǡ ߲௞ ࢞ሻ ൌ ሾ߲௜ ሺ߲௝ ࢞ǡ ߲௞ ࢞ሻ ൅ ߲௝ ሺ߲௜ ࢞ǡ ߲௞ ࢞ሻ െ ߲௞ ሺ߲௜ ࢞ǡ ߲௝ ࢞ሻሿ ʹ is true for all ݅ǡ ݆ǡ ݇ ൌ ͳǡʹǡ Ǥ Ǥ Ǥ ǡ ݉. In particular it implies that every entry of ߲௜ ܺܺ ௧ is a differential rational function of ሺ߲௜ ࢞ǡ ߲௝ ࢞ሻଵஸ௜ஸ௝ஸ௡ ǡ ሺ߲௜ ࢞ǡ ߲௝ ࢞ሻ௡ାଵஸ௜ஸ௠ǡଵஸ௝ஸ௡ . In b) case a justification similar to a) case shows that whenever ݂ሺܼሻ ‫ܥ א‬ሺܼሻௌைሺ௡ǡ஼ሻ then ݂ሺܼሻ will be in ‫ܥ‬ሺሺࢠ௜ ǡ ࢠ௝ ሻଵஸ௜ஸ௝ஸ௡ ǡ ሺ ܼሻሻሺඥሺ‫ܨ‬ଵ ሺܼሻǡ ‫ܨ‬ଵ ሺܼሻሻǡ ඥሺ‫ܨ‬ଶ ሺܼሻǡ ‫ܨ‬ଶ ሺܼሻሻǡ Ǥ Ǥ Ǥ ǡ ඥሺ‫ܨ‬௡ ሺܼሻǡ ‫ܨ‬௡ ሺܼሻሻሻ that is ݂ሺܼሻ ‫ܥ א‬ሺሺࢠ௜ ǡ ࢠ௝ ሻଵஸ௜ஸ௝ஸ௡ ǡ ܼሻ. As to the inequality ݅ ൅ ݆ ൏ ʹ݊ appearing in the system of generates in b) case it is due the equality ೔ ೕ ሺ ሺ߲ ఈ ࢞ǡ ߲ ఈ ࢞ሻଵஸ௜ஸ௝ஸ௡ ሻ ൌ ሺ ܺሻଶ ೙

೙

This equality enables one to drop from the system of generators in a) the element ሺ߲ ఈ ࢞ǡ ߲ ఈ ࢞ሻ. In ‫ ܥ‬ൌ ܴ, ݊ ൌ ݉ case of this result can be found in [8].

SEPARATING SYSTEMS OF INVARIANTS, UNIQUENESS AND EXISTENCE In future ܸ଴ stands for the set ܸ଴ ൌ ሼ࢛ ൌ ሺ‫ݑ‬ଵ ǡ ‫ݑ‬ଶ ǡ Ǥ Ǥ Ǥ ǡ ‫ݑ‬௡ ሻ ‫ ܨ א‬௡ ǣ ܺሺ࢛ሻ᩸ሽ which is an ‫ܮܩ‬ሺ݊ǡ ‫ܥ‬ሻ ‫ ܥ ڈ‬௡ - invariant subset of ‫ ܨ‬௡ and it is assumed that it is not empty. Let ‫כ‬ǣ ‫ܮܩ‬ሺ݊ǡ ‫ܨ‬ሻ ՜ ‫ܮܩ‬ሺ݊ǡ ‫ܨ‬ሻ be any group anti-homomorphism, that is ሺ݃ଵ ݃ଶ ሻ‫ כ‬ൌ ݃ଶ‫݃ כ‬ଵ‫ כ‬for any ݃ଵ ǡ ݃ଶ ‫א‬ ‫ܮܩ‬ሺ݊ǡ ‫ܨ‬ሻ, ‫ ܧ‬stand for any ݊-order square matrix over ‫ܨ‬. Consider the following subgroups of the Affine group ‫ܪ‬଴ா ൌ ሼ݄ ‫ܮܩ א‬ሺ݊ǡ ‫ܥ‬ሻǣ݄‫ כ݄ܧ‬ൌ ‫ܧ‬ሽǡ ܵ‫ܪ‬଴ா ൌ ‫ܪ‬଴ா ‫ܮܵ ת‬ሺ݊ǡ ‫ ܥ‬௡ ሻǡ ‫ܪ‬ா ൌ ‫ܪ‬଴ா ‫ ܥ ڈ‬௡ ǡ ܵ‫ܪ‬ா ൌ ܵ‫ܪ‬଴ா ‫ ܥ ڈ‬௡ Note that all classic subgroups of ‫ܮܩ‬ሺ݊ǡ ‫ܥ‬ሻ are in the form ‫ܪ‬଴ா or ܵ‫ܪ‬଴ா . For example, it is easy to see that if ‫כ‬ǣ ݃ հ ݃ିଵ , ‫ ܧ‬the identity matrix then ‫ܪ‬଴ா is ‫ܮܩ‬ሺ݊ǡ ‫ܥ‬ሻ and ܵ‫ܪ‬଴ா ൌ ܵ‫ܮ‬ሺ݊ǡ ‫ܥ‬ሻ, if ‫כ‬ǣ ݃ հ ݃௧ , ‫ ܧ‬the identitymatrix then ‫ܪ‬଴ா ൌ ܱሺ݊ǡ ‫ܥ‬ሻ and ܵ‫ܪ‬଴ா ൌ ܱܵሺ݊ǡ ‫ܥ‬ሻ. Theorem 3.1: a) If ݊ ൐ ݉ (݊ ൑ ݉) then elements ࢛ǡ ࢜ ‫ܸ א‬଴ are ‫ܪ‬ா - equivalent, that is ࢜ ൌ ࢛݄ ൅ ݄଴ for some ሺ݄ǡ ݄଴ ሻ ‫ܪ א‬ா , if and only if for them the following equalities ሺ߲௜ ܺሺ࢛ሻܺ ିଵ ሺ࢛ሻ ൌ ߲௜ ܺሺ࢜ሻܺ ିଵ ሺ࢜ሻሻଵஸ௜ஸ௠ ܺሺ࢛ሻ‫ כ ܺܧ‬ሺ࢛ሻ ൌ ܺሺ࢜ሻ‫ כ ܺܧ‬ሺ࢜ሻ


ሺǤ ሺ߲௜ ܺሺ࢛ሻܺ ିଵ ሺ࢛ሻ ൌ ߲௜ ܺሺ࢜ሻܺ ିଵ ሺ࢜ሻሻଵஸ௜ஸ௠ ǡ ሺ߲௝ ࢛ܺ ିଵ ሺ࢛ሻ ൌ ߲௝ ࢜ܺ ିଵ ሺ࢜ሻሻ௡ାଵஸ௝ஸ௠ ܺሺ࢛ሻ‫ כ ܺܧ‬ሺ࢛ሻ ൌ ܺሺ࢜ሻ‫ כ ܺܧ‬ሺ࢜ሻሻ are true. b) If ݊ ൐ ݉ (݊ ൑ ݉) then elements ࢛ǡ ࢜ ‫ܸ א‬଴ are ܵ‫ܪ‬ா - equivalent if and only if for them the following equalities ሺ߲௜ ܺሺ࢛ሻܺ ିଵ ሺ࢛ሻ ൌ ߲௜ ܺሺ࢜ሻܺ ିଵ ሺ࢜ሻሻଵஸ௜ஸ௠ ǡ ܺሺ࢛ሻ‫ כ ܺܧ‬ሺ࢛ሻ ൌ ܺሺ࢜ሻ‫ כ ܺܧ‬ሺ࢜ሻ ܺ ሺ࢛ሻ ൌ ܺ ሺ࢜ሻ ሺǤ ሺ߲௜ ܺሺ࢛ሻܺ ିଵ ሺ࢛ሻ ൌ ߲௜ ܺሺ࢜ሻܺ ିଵ ሺ࢜ሻሻଵஸ௜ஸ௠ ǡ ሺ߲௝ ࢛ܺ ିଵ ሺ࢛ሻ ൌ ߲௝ ࢜ܺ ିଵ ሺ࢜ሻሻ௡ାଵஸ௝ஸ௠ ǡ ܺሺ࢛ሻ‫ כ ܺܧ‬ሺ࢛ሻ ൌ ܺሺ࢜ሻ‫ כ ܺܧ‬ሺ࢜ሻ ܺ ሺ࢛ሻ ൌ ܺ ሺ࢜ሻሻ are true. Proof: If ࢛ǡ ࢜ ‫ܸ א‬଴ are equivalent with respect to the corresponding motion group then necessity of the corresponding equalities is evident. Therefore it is enough to show sufficiency. First of all note that equalities ሺ߲௜ ܺሺ࢛ሻܺ ିଵ ሺ࢛ሻ ൌ ߲௜ ܺሺ࢜ሻܺ ିଵ ሺ࢜ሻሻଵஸ௜ஸ௠ can be written as ߲௜ ሺܺ ିଵ ሺ࢛ሻܺሺ࢜ሻሻ ൌ Ͳǡ ݅ ൌ ͳǡʹǡ Ǥ Ǥ Ǥ ǡ ݉ , which means that ݄ ൌ ܺ ିଵ ሺ࢛ሻሻܺሺ࢜ሻ ‫ܮܩ א‬ሺ݊ǡ ‫ܥ‬ሻ that is ܺሺ࢜ሻ ൌ ܺሺ࢛ሻ݄, ܺሺ࢜ െ ࢛݄ሻ ൌ Ͳ for some ݄ ‫א‬ ‫ܮܩ‬ሺ݊ǡ ‫ܥ‬ሻ. Therefore for both cases of this theorem due to the equality ܺሺ࢛ሻ‫ כ ܺܧ‬ሺ࢛ሻ ൌ ܺሺ࢜ሻ‫ כ ܺܧ‬ሺ࢜ሻ one has ݄‫ כ݄ܧ‬ൌ ‫ܧ‬. Now we show that ‫ ݒ‬െ ‫ ܥ א ݄ݑ‬௡ . It can be done for both parts of this theorem in the same following way: If ݊ ൐ ݉ the equality ܺሺ࢜ െ ࢛݄ሻ ൌ Ͳ implies that ߲௜ ሺ࢜ െ ࢛݄ሻ ൌ Ͳ for all ݅ ൌ ͳǡʹǡ Ǥ Ǥ Ǥ ǡ ݉. Therefore in this case ࢜ െ ࢛݄ ൌ ݄଴ ‫ ܥ א‬௡ , that is the elements ࢛ǡ ࢜ are ‫ܪ‬ா - equivalent. If ݊ ൑ ݉ then ܺሺ࢜ሻ ൌ ܺሺ࢛ሻ݄ implies ߲௜ ሺ࢜ െ ࢛݄ሻ ൌ Ͳ only for all ݅ ൌ ͳǡʹǡ Ǥ Ǥ Ǥ ǡ ݊. But ሺ߲௝ ࢛ܺ ିଵ ሺ࢛ሻ ൌ ߲௝ ࢜ܺ ିଵ ሺ࢜ሻሻ௡ାଵஸ௝ஸ௠ implies that ߲௝ ࢛݄ ൌ ߲௝ ࢜ǡ ݆ ൌ ݊ ൅ ͳǡ Ǥ Ǥ Ǥ ǡ ݉ and therefore ߲௝ ሺ࢜ െ ࢛݄ሻ ൌ Ͳ, as far as ݄ ‫ܮܩ א‬ሺ݊ǡ ‫ܥ‬ሻ, for all ݆ ൌ ݊ ൅ ͳǡ Ǥ Ǥ Ǥ ǡ ݉ as well, that is once again one has ݄଴ ൌ ࢜ െ ࢛݄ ‫ ܥ א‬௡ , elements ࢛ǡ ࢜ are ‫ܪ‬ா - equivalent. In case of b) due to the equality ܺ ሺ࢛ሻ ൌ ܺ ሺ࢜ሻ one can conclude that ݄ ൌ ͳ that is ሺ݄ǡ ݄଴ ሻ ‫א‬ ܵ‫ܪ‬ா , elements ࢛ǡ ࢜ are ܵ‫ܪ‬ா - equivalent. This theorem provides complete ([8]) systems of invariants for groups ‫ܪ‬ா , ܵ‫ܪ‬ா and, in particular, shows that presented in Theorem 2.2 systems of generators are complete as well. Remark 3.1: If one proves that the presented separating system of differential invariants in Theorem 3.1 generates the corresponding ߲-differential fields over ‫ ܥ‬then it will be a generalization of Theorems 2.2 and 2.3. For the time being what we can conclude from Theorem 3.1 is that each element of the corresponding differential field is a "function" of the corresponding separating system of invariants. Now let us discuss the permissible values of ߲௜ ܺሺ࢛ሻܺ ିଵ ሺ࢛ሻ and ܺሺ࢛ሻ‫ כ ܺܧ‬ሺ࢛ሻ. That is, for example, for which system of ݊-order square matrices ሺ‫ܣ‬௜ ሻ௜ୀଵǡǤǤǤǡ௠ ǡ ‫ ܤ‬over ‫ ܨ‬one can find such ࢛ ‫ ܨ א‬௡ that ሺ߲௜ ܺሺ࢛ሻܺ ିଵ ሺ࢛ሻ ൌ ‫ܣ‬௜ ሻଵஸ௜ஸ௠ ܺሺ࢛ሻ‫ כ ܺܧ‬ሺ࢛ሻ ൌ ‫ܤ‬ǫ Here we are going to give an answer to this question at least in functional field ‫ ܨ‬case, that is when ‫ ܥ‬is the field of real or complex numbers and ‫ ܨ‬is a differential field of functions of ࢚ ൌ ሺ‫ݐ‬ଵ ǡ ‫ݐ‬ଶ ǡ Ǥ Ǥ Ǥ ǡ ‫ݐ‬௠ ሻ with values in ‫ܥ‬, డ ߲௜ ൌ . But for the time being let ‫ ܨ‬be any differential field and ‫ ܪ‬ൌ ‫ܪ‬ா . For any invertible matrix ܻ one has డ௧೔

߲௝ ሺ߲௜ ܻܻ ିଵ ሻ െ ߲௜ ሺ߲௝ ܻܻ ିଵ ሻ ൌ ߲௝ ܻܻ ିଵ ߲௜ ܻܻ ିଵ െ ߲௜ ܻܻ ିଵ ߲௝ ܻܻ ିଵ Therefore for solvability of the system the equalities

ሺ߲௜ ܻܻ ିଵ ൌ ‫ܣ‬௜ ሻଵஸ௜ஸ௠

(2)

ሺ߲௜ ‫ܣ‬௝ െ ߲௝ ‫ܣ‬௜ ൌ ‫ܣ‬௝ ‫ܣ‬௜ െ ‫ܣ‬௜ ‫ܣ‬௝ ሻଵஸ௜ழ௝ஸ௠ are necessary. These equalities are known as the "Integrability condition" for the above system. It is known [9] that in algebraic closed constant field ‫ ܥ‬case this "Integrability condition" is enough for solvability of system (2) in some differential extension of ‫ ܨ‬with the same constant field ‫ܥ‬. In future it will be assumed that ‫ ܨ‬is rich enough to make the system solvable in ‫ ܨ‬itself.


If one assumes that entries of ‫ ܧ‬are in ‫ܥ‬, ‫ כ‬commutes with ߲ଵ ǡ ߲ଶ ǡ ‫ ڮ‬ǡ ߲௠ and taking inverse, that is ሺ݃ିଵ ሻ‫ כ‬ൌ ሺ݃‫ כ‬ሻିଵ , then due to ߲௜ ሺܻ‫ כ ܻܧ‬ሻ ൌ ߲௜ ܻܻ ିଵ ሺܻ‫ כ ܻܧ‬ሻ ൅ ܻ‫ כ ܻܧ‬ሺ߲௜ ܻܻ ିଵ ሻ‫כ‬ for solvability of the system ሺ߲௜ ܻܻ ିଵ ൌ ‫ܣ‬௜ ሻଵஸ௜ஸ௠ ǡ ܻ‫ כ ܻܧ‬ൌ ‫ܤ‬ the equalities ሺ߲௜ ‫ܣ‬௝ െ ߲௝ ‫ܣ‬௜ ൌ ‫ܣ‬௝ ‫ܣ‬௜ െ ‫ܣ‬௜ ‫ܣ‬௝ ሻଵஸ௜ழ௝ஸ௠ ǡ ሺ߲௜ ‫ ܤ‬ൌ ‫ܣ‬௜ ‫ ܤ‬െ ‫כܣܤ‬௜ ሻଵஸ௜ஸ௠

(3)

are necessary. But these equalities are not sufficient. They may guarantee only existence of ܻ and such ‫ܧ‬଴ with constant entries for which ሺ߲௜ ܻܻ ିଵ ൌ ‫ܣ‬௜ ሻଵஸ௜ஸ௠ ǡ ܻ‫ܧ‬଴ ܻ ‫ כ‬ൌ ‫ܤ‬ are true. Indeed for any solution ܻ଴ of system (3) it is easy to check that ߲௜ ሺܻ଴ିଵ ‫ܤ‬ሺܻ଴‫ כ‬ሻିଵ ሻ ൌ Ͳ for any ݅ ൌ ͳǡʹǡ ‫ ڮ‬ǡ ݉ that is ܻ଴ିଵ ‫ܤ‬ሺܻ଴‫ כ‬ሻିଵ ൌ ‫ܧ‬଴ , ‫ ܤ‬ൌ ܻ଴ ‫ܧ‬଴ ܻ଴‫ כ‬. But in an abstract differential field case which condition enables one to be sure that ‫ܧ‬଴ ൌ ‫ ?ܧ‬We don’t know and because of it we are not able to consider this problem here in pure abstract case. In functional differential field case an integrability condition for (3) can be given in the following form ሺ߲௜ ‫ܣ‬௝ െ ߲௝ ‫ܣ‬௜ ൌ ‫ܣ‬௝ ‫ܣ‬௜ െ ‫ܣ‬௜ ‫ܣ‬௝ ሻଵஸ௜ழ௝ஸ௠ ǡ ሺ߲௜ ‫ ܤ‬ൌ ‫ܣ‬௜ ‫ ܤ‬െ ‫כܣܤ‬௜ ሻଵஸ௜ஸ௠ ǡ ‫ܤ‬ሺͲሻ ൌ ݃‫ כ݃ܧ‬᩸݃ ‫ܮܩ א‬ሺ݊ǡ ‫ܥ‬ሻǡ where ‫ܤ‬ሺͲሻ stands for the value of ‫ ܤ‬at ࢚ ൌ Ͳ. To see it consider the same solution ܻ଴ of system (3). Due to ‫ ܤ‬ൌ ܻ଴ ‫ܧ‬଴ ܻ଴‫ כ‬ǡ ‫ܤ‬ሺͲሻ ൌ ݃‫כ݃ܧ‬ one has ‫ܧ‬଴ ൌ ܻ଴ିଵ ሺͲሻ݃‫ܧ‬ሺܻ଴ିଵ ሺͲሻሻ‫ כ‬that is ‫ ܤ‬ൌ ሺܻ଴ ܻ଴ିଵ ሺͲሻ݃ሻ‫ܧ‬ሺܻ଴ ܻ଴ିଵ ሺͲሻ݃ሻ‫כ‬ and ܻ ൌ ܻ଴ ܻ଴ିଵ ሺͲሻ݃ is also solution for system (3). Now let us consider ‫ ܪ‬ൌ ܵ‫ܪ‬ா case. In this case we assume that if ‫ ܣ‬ൌ ‫ ܤ‬then ‫ כܣ‬ൌ ‫ כ ܤ‬. For nonzero ܽ ‫ ܨ א‬we define ܽ‫ כ‬to be ‫ כܣ‬, where ‫ ܣ‬ൌ ܽ. In future it is assumed that every constant ܿ for which ܿܿ ‫ כ‬ൌ ͳ is representable as ܿ ൌ ݄ for some ݄ ‫ܪ א‬଴ா and the matrix ‫ ܧ‬is invertible. Note that if ܻ is any solution of (2) then for ܾ ൌ ܻ one has ߲௜ ܾܾ ିଵ ൌ ሺ‫ܣ‬௜ ሻ for any ݅ ൌ ͳǡʹǡ ‫ ڮ‬ǡ ݉. Therefore the equalities ߲௜ ሺ‫ܣ‬௝ ሻ ൌ ߲௝ ሺ‫ܣ‬௜ ሻ are true, where stands for the trace. Moreover if ܻ‫ כ ܻܧ‬ൌ ‫ ܤ‬then ‫ ܤ‬ൌ ܾܾ ‫ܧ כ‬. Now assume that ሺ߲௜ ‫ܣ‬௝ െ ߲௝ ‫ܣ‬௜ ൌ ‫ܣ‬௝ ‫ܣ‬௜ െ ‫ܣ‬௜ ‫ܣ‬௝ ሻଵஸ௜ழ௝ஸ௠ ǡ ሺ߲௜ ‫ ܤ‬ൌ ‫ܣ‬௜ ‫ ܤ‬െ ‫כܣܤ‬௜ ሻଵஸ௜ஸ௠ ǡ ‫ܤ‬ሺͲሻ ൌ ݃‫ כ݃ܧ‬᩸݃ ‫ܮܩ א‬ሺ݊ǡ ‫ܥ‬ሻǡ ‫ ܤ‬ൌ ܾܾ ‫ ܧ כ‬ǡ ሺ߲௜ ܾܾ ିଵ ൌ ሺ‫ܣ‬௜ ሻሻଵஸ௜ஸ௠ Under these conditions the following system ሺ߲௜ ܻܻ ିଵ ൌ ‫ܣ‬௜ ሻ௜ୀଵǡଶǡ‫ڮ‬ǡ௠ ǡ ܻ‫ כ ܻܧ‬ൌ ‫ܤ‬ǡ ܻ ൌ ܾ

(4)

has solution. Indeed let ܻ଴ be any solution of (3). As we have noted ܻ଴ is a solution for the system ሺ߲௜ ‫ି ݕݕ‬ଵ ൌ ሺ‫ܣ‬௜ ሻሻଵஸ௜ஸ௠ , ܻ଴ ܾ ିଵ ൌ ܿ is a constant, ܿܿ ‫ כ‬ൌ ͳ due to ܻ଴ ‫ܻܧ‬଴‫ כ‬ൌ ‫ ܤ‬and therefore ܻ଴ ܾ ିଵ ൌ ݄ for some ݄ ‫ܪ א‬଴ா . So ܻ ൌ ܻ଴ ݄ିଵ satisfies (4). భ మ ೙ But if ݊ ൐ ݉ for ‫ ܪ‬ൌ ‫ܪ‬ா (‫ ܪ‬ൌ ܵ‫ܪ‬ா ) case ܻ should be in the form ሾ߲ ఈ ࢛ǡ ߲ ఈ ࢛ǡ ‫ ڮ‬ǡ ߲ ఈ ࢛ሿ which means that the following identities ೔ ೕ ሺ߲ ఈ ܻ ఈ ൌ ߲ఉ ܻ ఈ ሻఈାఈ೔ୀఉାఈೕ


೔

should hold, where ܻ ఈ stands for the corresponding to ߙ ௜ row of ܻ. Note that all these identities can be written ೔ ೕ ೔ ೕ in terms of matrices ሺ‫ܣ‬௜ ሻ௜ୀଵǡଶǡ‫ڮ‬ǡ௠ over ‫ܥ‬. For example, if ߲௞ ܻ ఈ ൌ ߲௟ ܻ ఈ it can be written as ‫ܣ‬ఈ௞ ܻ ൌ ‫ܣ‬ఈ௟ ܻ that ೔

ೕ

is ‫ܣ‬ఈ௞ ൌ ‫ܣ‬ఈ௟ . Moreover the system of identities can be given by the finite system of identities ೔

ೕ

ሺ߲ ఈ ܻ ఈ ൌ ߲ఉ ܻ ఈ ሻఈାఈ೔ୀఉାఈೕ , where for any ݈ ൌ ͳǡʹǡ ‫ ڮ‬ǡ ݉, Ͳ ൑ ߙ௟ ൑ ௞ ߙ௟௞ , Ͳ ൑ ߚ௟ ൑ ௞ ߙ௟௞ , ߙ௟ ߚ௟ ൌ Ͳ. So in functional differential field case if for entries of matrices ሺ‫ܣ‬௜ ሺ࢚ሻሻ௜ୀଵǡଶǡ‫ڮ‬ǡ௠ all these equalities also hold ೖ then one should look for solution of the system ߲ ௞ ࢟ ൌ ܻ଴ఈ at any ݇ ൌ ͳǡʹǡ ‫ ڮ‬ǡ ݉, where ܻ଴ is any particular solution of the system ሺ͵ሻ (respect. ሺͶሻ). భ మ ೙ If ࢟ ൌ ࢞ሺ࢚ሻ is any solution of it then for ܺ ൌ ሾ߲ ఈ ࢞ǡ ߲ ఈ ࢞ǡ ‫ ڮ‬ǡ ߲ ఈ ࢞ሿ one has ሺ߲௜ ܺܺ ିଵ ൌ ‫ܣ‬௜ ሻଵஸ௜ஸ௠ ǡ ܺ‫ כ ܺܧ‬ൌ ‫ܤ‬ ሺǤ ሺ߲௜ ܺܺ ିଵ ൌ ‫ܣ‬௜ ሻଵஸ௜ஸ௠ ǡ ܺ‫ כ ܺܧ‬ൌ ‫ܤ‬ǡ ܺ ൌ ܾሻ In ݊ ൑ ݉, ‫ ܪ‬ൌ ‫ܪ‬ா (‫ ܪ‬ൌ ܵ‫ܪ‬ா ) case one has to consider solvability of the system (respect.

ሺ߲௜ ܺሺ࢞ሻܺሺ࢞ሻିଵ ൌ ‫ܣ‬௜ ሻଵஸ௜ஸ௠ ǡ ܺሺ࢞ሻ‫ כ ܺܧ‬ሺ࢞ሻ ൌ ‫ܤ‬ǡ ሺ߲௝ ࢞ܺሺ࢞ሻିଵ ൌ ࢇ௝ ሻ௡ାଵஸ௝ஸ௠

(5)

ሺ߲௜ ܺሺ࢞ሻܺሺ࢞ሻିଵ ൌ ‫ܣ‬௜ ሻଵஸ௜ஸ௠ ǡ ܺሺ࢞ሻ‫ כ ܺܧ‬ሺ࢞ሻ ൌ ‫ܤ‬ǡ ሺ߲௝ ࢞ܺሺ࢞ሻିଵ ൌ ࢇ௝ ሻ௡ାଵஸ௝ஸ௠ ǡ ܺ ሺ࢞ሻ ൌ ܾሻ

(6)

In this case necessity of the following equalities ሺ߲௜ ‫ܣ‬௝ െ ߲௝ ‫ܣ‬௜ ൌ ‫ܣ‬௝ ‫ܣ‬௜ െ ‫ܣ‬௜ ‫ܣ‬௝ ሻଵஸ௜ழ௝ஸ௠ ǡ ሺ߲௞ ࢇ௝ െ ߲௝ ࢇ௞ ൌ ࢇ௞ ‫ܣ‬௝ െ ࢇ௝ ‫ܣ‬௞ ሻ௡ାଵஸ௞ழ௝ஸ௠ ǡ

(7)

೔

ሺǤ ሺ߲௜ ࢇ௝ ൌ ‫ܣ‬௝ఈ െ ࢇ௝ ‫ܣ‬௜ ሻଵஸ௜ழ௡ାଵஸ௝ஸ௠ ǡ ሺ߲௜ ‫ ܤ‬ൌ ‫ܣ‬௜ ‫ ܤ‬െ ‫כܣܤ‬௜ ሻଵஸ௜ஸ௠ ǡ ‫ ܤ‬ൌ ܾܾ ‫ ܧ כ‬ǡ ሺ߲௜ ܾܾ ିଵ ൌ ሺ‫ܣ‬௜ ሻሻଵஸ௜ஸ௠ ሻ

(8)

can be checked easily. Here the equalities ೔

ሺ߲௞ ࢇ௝ െ ߲௝ ࢇ௞ ൌ ࢇ௝ ‫ܣ‬௞ െ ࢇ௞ ‫ܣ‬௝ ሻ௡ାଵஸ௞ழ௝ஸ௠ ǡ ሺ߲௜ ࢇ௝ ൌ ‫ܣ‬௝ఈ െ ࢇ௝ ‫ܣ‬௜ ሻଵஸ௜ழ௡ାଵஸ௝ஸ௠ should be required because of ߲௜ ߲௝ ࢞ ൌ ߲௝ ߲௜ ࢞. In functional field case one more equality, namely ‫ܤ‬ሺͲሻ ൌ ݃‫ כ݃ܧ‬should be added to ሺ͹ሻ (respect. ሺͺሻ) to have sufficient condition for integrability of ሺͷሻ (respect. ሺ͸ሻ). In this case it can be solved in the following way: 1. Solve the system ሺ߲௜ ܻܻ ିଵ ൌ ‫ܣ‬௜ ሻଵஸ௜ஸ௠ ǡ ܻ‫ כ ܻܧ‬ൌ ‫ܤ‬ǡ ‫ܤ‬ሺͲሻ ൌ ݃‫כ݃ܧ‬ ሺǤ ሺ߲௜ ܻܻ ିଵ ൌ ‫ܣ‬௜ ሻଵஸ௜ஸ௠ ǡ ܻ‫ כ ܻܧ‬ൌ ‫ܤ‬ǡ ‫ܤ‬ሺͲሻ ൌ ݃‫ כ݃ܧ‬ǡ ܻ ൌ ܾሻ 2. Consider its any particular solution ܻ଴ ሺ࢚ሻ. 3. Solve the system: ೔ ሺ߲ ௜ ࢟ ൌ ܻ଴ఈ ሻଵஸ௜ஸ௡ ǡ ሺ߲ ௝ ࢟ ൌ ࢇ௝ ܻ଴ ሻ௡ାଵஸ௝ஸ௠ If ࢟ ൌ ࢞ሺ࢚ሻ is any its solution then for ܺ ൌ ܺሺ࢞ሺ࢚ሻሻ the system of equalities (5) (respect. (6)) is valid. So in functional differential field ‫ ܨ‬case we can state the following Gauss-Codazzi type theorem for the groups ‫ܪ‬ா and ܵ‫ܪ‬ா . In this Theorem it is assumed that entries of ‫ ܧ‬are in ‫ܥ‬, ‫ כ‬commutes with ߲ଵ ǡ ߲ଶ ǡ ‫ ڮ‬ǡ ߲௠ , ሺ݃ିଵ ሻ‫ כ‬ൌ ሺ݃‫ כ‬ሻିଵ for any ݃ ‫ܮܩ א‬ሺ݊ǡ ‫ܥ‬ሻ, ݃ଵ‫ כ‬ൌ ݃ଶ‫ כ‬whenever ݃ଵ ൌ ݃ଶ and in ‫ ܪ‬ൌ ܵ‫ܪ‬ா case the matrix ‫ ܧ‬is invertible. Theorem 3.2: a) If ݊ ൐ ݉ and ሺ‫ܣ‬௜ ሻ௜ୀଵǡଶǡǤǤǤǡ௠ ǡ ‫ ܤ‬are such ݊-order square matrices over ‫ ( ܨ‬and ܾ ‫ )ܨ א‬for which the following equalities ሺ߲௜ ‫ܣ‬௝ െ ߲௝ ‫ܣ‬௜ ൌ ‫ܣ‬௝ ‫ܣ‬௜ െ ‫ܣ‬௜ ‫ܣ‬௝ ሻଵஸ௜ழ௝ஸ௠ ǡ ሺ߲௜ ‫ ܤ‬ൌ ‫ܣ‬௜ ‫ ܤ‬െ ‫כܣܤ‬௜ ሻଵஸ௜ஸ௠ ǡ ‫ܤ‬ሺͲሻ ൌ ݃‫ כ݃ܧ‬᩸݃ ‫ܮܩ א‬ሺ݊ǡ ‫ܥ‬ሻ


ሺǤ ሺ߲௜ ‫ܣ‬௝ െ ߲௝ ‫ܣ‬௜ ൌ ‫ܣ‬௝ ‫ܣ‬௜ െ ‫ܣ‬௜ ‫ܣ‬௝ ሻଵஸ௜ழ௝ஸ௠ ǡ ሺ߲௜ ‫ ܤ‬ൌ ‫ܣ‬௜ ‫ ܤ‬െ ‫כܣܤ‬௜ ሻଵஸ௜ஸ௠ ǡ ‫ܤ‬ሺͲሻ ൌ ݃‫כ݃ܧ‬ ᩸݃ ‫ܮܩ א‬ሺ݊ǡ ‫ܥ‬ሻǡ ‫ ܤ‬ൌ ܾܾ ‫ ܧ כ‬ǡ ሺ߲௜ ܾܾ ିଵ ൌ ሺ‫ܣ‬௜ ሻሻଵஸ௜ஸ௠ ሻ are true then there exists ࢛ ‫ ܨ א‬௡ , unique up to ‫ܪ‬ா (respect. ܵ‫ܪ‬ா ) equivalence, for which ሺ߲௜ ܺሺ࢛ሻܺ ିଵ ሺ࢛ሻ ൌ ‫ܣ‬௜ ሻଵஸ௜ஸ௠ ܺሺ࢛ሻ‫ כ ܺܧ‬ሺ࢛ሻ ൌ ‫ܤ‬ ሺǤ ሺ߲௜ ܺሺ࢛ሻܺ ିଵ ሺ࢛ሻ ൌ ‫ܣ‬௜ ሻଵஸ௜ஸ௠ ǡ ܺሺ࢛ሻ‫ כ ܺܧ‬ሺ࢛ሻ ൌ ‫ ܺ ܤ‬ሺ࢛ሻ ൌ ܾሻ b) If ݊ ൑ ݉ and ሺ‫ܣ‬௜ ሻଵஸ௜ஸ௠ ǡ ‫ ܤ‬are such ݊-order square matrices, ሺࢇ௝ ሻ௡ାଵஸ௝ஸ௠ are row vectors over ‫ܨ‬ ( and ܾ ‫ )ܨ א‬for which the following equalities ሺ߲௜ ‫ܣ‬௝ െ ߲௝ ‫ܣ‬௜ ൌ ‫ܣ‬௝ ‫ܣ‬௜ െ ‫ܣ‬௜ ‫ܣ‬௝ ሻଵஸ௜ழ௝ஸ௠ ǡ ሺ߲௜ ‫ ܤ‬ൌ ‫ܣ‬௜ ‫ ܤ‬െ ‫כܣܤ‬௜ ሻଵஸ௜ஸ௠ ǡ ‫ܤ‬ሺͲሻ ൌ ݃‫ כ݃ܧ‬᩸ ೔

݃ ‫ܮܩ א‬ሺ݊ǡ ‫ܥ‬ሻǡ ሺ߲௞ ࢇ௝ െ ߲௝ ࢇ௞ ൌ ࢇ௞ ‫ܣ‬௝ െ ࢇ௝ ‫ܣ‬௞ ሻ௡ାଵஸ௞ழ௝ஸ௠ ǡ ሺ߲௜ ࢇ௝ ൌ ‫ܣ‬௝ఈ െ ࢇ௝ ‫ܣ‬௜ ሻଵஸ௜ழ௡ାଵஸ௝ஸ௠ ሺǤ ሺ߲௜ ‫ܣ‬௝ െ ߲௝ ‫ܣ‬௜ ൌ ‫ܣ‬௝ ‫ܣ‬௜ െ ‫ܣ‬௜ ‫ܣ‬௝ ሻଵஸ௜ழ௝ஸ௠ ǡ ሺ߲௜ ‫ ܤ‬ൌ ‫ܣ‬௜ ‫ ܤ‬െ ‫כܣܤ‬௜ ሻଵஸ௜ஸ௠ ǡ ‫ܤ‬ሺͲሻ ൌ ݃‫כ݃ܧ‬ ᩸݃ ‫ܮܩ א‬ሺ݊ǡ ‫ܥ‬ሻǡ ሺ߲௞ ࢇ௝ െ ߲௝ ࢇ௞ ൌ ࢇ௞ ‫ܣ‬௝ െ ࢇ௝ ‫ܣ‬௞ ሻ௡ାଵஸ௞ழ௝ஸ௠ ǡ ೔

ሺ߲௜ ࢇ௝ ൌ ‫ܣ‬௝ఈ െ ࢇ௝ ‫ܣ‬௜ ሻଵஸ௜ழ௡ାଵஸ௝ஸ௠ ǡ ‫ ܤ‬ൌ ܾܾ ‫ ܧ כ‬ǡ ሺ߲௜ ܾܾ ିଵ ൌ ሺ‫ܣ‬௜ ሻሻଵஸ௜ஸ௠ ሻ are true then there exists ࢛ ‫ ܨ א‬௡ , unique up to ‫ܪ‬ா (respect. ܵ‫ܪ‬ா ) equivalence, for which ሺ߲௜ ܺሺ࢛ሻܺ ିଵ ሺ࢛ሻ ൌ ‫ܣ‬௜ ሻଵஸ௜ஸ௠ ǡ ܺሺ࢛ሻ‫ כ ܺܧ‬ሺ࢛ሻ ൌ ‫ܤ‬ሺ߲௝ ࢛ܺ ିଵ ሺ࢛ሻ ൌ ࢇ௝ ሻ௡ାଵஸ௝ஸ௠ ሺǤ ሺ߲௜ ܺሺ࢛ሻܺ ିଵ ሺ࢛ሻ ൌ ‫ܣ‬௜ ሻଵஸ௜ஸ௠ ǡ ܺሺ࢛ሻ‫ כ ܺܧ‬ሺ࢛ሻ ൌ ‫ܤ‬ǡ ሺ߲௝ ࢛ܺ ିଵ ሺ࢛ሻ ൌ ࢇ௝ ሻ௡ାଵஸ௝ஸ௠ ܺ ሺ࢛ሻ ൌ ܾሻ

ACKNOWLEDGMENTS This research is supported by IIUM grant END B13-033-0918.

REFERENCES 1 2. 3. 4. 5. 6. 7. 8. 9.

H. Weyl H., The Classical Groups, Their Invariants and Representations, Princeton Academic Press, 1997, pp. 1-320. H. Kraft and C. Procesi, Classical Invariant Theory. A Primer, 1996, pp.1-123. U. D. Bekbaev, Bulletin of Malaysian Mathematical Society 28, N1, 55-60 (2005). U. Bekbaev. On the field of differential rational invariants of a subgroup of affine group (Ordinary differential case). http://front.math.ucdavis.edu/, arXiv:math.AG/0608479. Pp,1-10. U. Bekbaev. On the field of differential rational invariants of a subgroup of affine group (Partial differential case). http://front.math.ucdavis.edu/, arXiv:math.AG/0609252. pp.1- 12. E. R. Kolchin, Differential Algebra and Algebraic Groups, New York: Academic Press, 1973, pp. 1-448. I. Kaplansky, An Introduction to Differential Algebra, Paris: Hermann, 1957. D. Khadjiev, Turk J. Math. 34, 543-559 (2010). M. Bronstein, Z. Li and M. Wu., “Picard-Vessiot Extensions for Linear Functional Systems”, ISSAC ACM Press, pp. 6875 (2005).
