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Generating systems of differential invariants and the theorem on existence for curves in the pseudo-Euclidean geometry

It is proved that the complete system of M(n,p)-invariant differential rational functions of a path (curve) is a generating system of the differential field of all M(n, p) -invariant differential rational functions of a path (curve), respectively. | Turkish Journal of Mathematics http://journals.tubitak.gov.tr/math/ Research Article Turk J Math (2013) 37: 80 – 94 ¨ ITAK ˙ c TUB doi:10.3906/mat-1104-41 Generating systems of differential invariants and the theorem on existence for curves in the pseudo-Euclidean geometry ∗ ¨ ˙ ¨ Djavvat KHADJIEV, Idris OREN , Omer PEKS ¸ EN Department of Mathematics, Karadeniz Technical University, Trabzon, 61080, Turkey Received: 27.04.2011 • Accepted: 09.08.2011 • Published Online: 17.12.2012 • Printed: 14.01.2013 Abstract: Let M (n, p) be the group of all motions of an n -dimensional pseudo-Euclidean space of index p. It is proved that the complete system of M(n,p)-invariant differential rational functions of a path (curve) is a generating system of the differential field of all M (n, p) -invariant differential rational functions of a path (curve), respectively. A fundamental system of relations between elements of the complete system of M(n,p)-invariant differential rational functions of a path (curve) is described. Key words: Curve, differential invariant, pseudo-Euclidean geometry, Minkowski geometry 1. Introduction The present paper is a continuation of our paper [18]. Let Epn be the n-dimensional pseudo-Euclidean space of index p (that is the space Rn with the scalar product = −x1 y1 − · · · − xp yp + xp+1 yp+1 + · · · + xn yn ), O(n, p) is the group of all pseudo-orthogonal transformations of Epn , M (n, p)={F : Epn → Epn | F x = gx + b , g ∈ O(n, p), b ∈ Epn } and SM (n, p) = {F ∈ M (n, p) : detg = 1} . Here, for groups G = M (n, p) and G = SM (n, p), we prove that the complete system of G-invariant differential rational functions of a path (curve) obtained in [18, Theorems 2–3 and Corollaries 1–2] is a generating system of the differential field of all G -invariant differential rational functions of a path (respectively, curve). We describe a fundamental system of relations between elements of the complete system of G-invariant functions of a path (curve) (i.e., .