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Translation ovoids of orthogonal polar spaces

Translation ovoids of orthogonal polar spaces Abstract. We prove that translation ovoids of a finite orthogonal polar space P exist if and only if P is one of Q þ ð3; qÞ, Qð4; qÞ, Q þ ð5; qÞ. 2000 Mathematics Subject Classification: 51E20. 1 Introduction Working in finite geometries or in algebraic geometry over finite fields, it is often very hard to determine whenever a system of polynomials over the Galois field GFðqÞ of order q has solutions and the following theorem, due to Chevalley and Warning, is very useful. Let q ¼ p e where p is a prime. If P1 ; P2 ; . . . ; Ps are polynomials of GFðqÞ½X1 ; X2 ; . . . ; Xn and di is the degree of Pi ði ¼ 1; 2; . . . ; sÞ, put d ¼ d1 þ d 2 þ Á Á Á þ d s and let N be the number of solutions over GFðqÞ of the system of polynomial equations: P1 ðX1 ; X2 ; . . . ; Xn Þ ¼ P2 ðX1 ; X2 ; . . . ; Xn Þ ¼ Á Á Á ¼ Ps ðX1 ; X2 ; . . . http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Forum Mathematicum de Gruyter

Translation ovoids of orthogonal polar spaces

LUNARDON, G.; POLVERINO, O.
Forum Mathematicum , Volume 16 (5) – Sep 16, 2004
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Publisher
de Gruyter
Copyright
Copyright © 2004 by Walter de Gruyter GmbH & Co. KG
ISSN
0933-7741
eISSN
1435-5337
DOI
10.1515/form.2004.029
Publisher site
See Article on Publisher Site

Abstract

Abstract. We prove that translation ovoids of a finite orthogonal polar space P exist if and only if P is one of Q þ ð3; qÞ, Qð4; qÞ, Q þ ð5; qÞ. 2000 Mathematics Subject Classification: 51E20. 1 Introduction Working in finite geometries or in algebraic geometry over finite fields, it is often very hard to determine whenever a system of polynomials over the Galois field GFðqÞ of order q has solutions and the following theorem, due to Chevalley and Warning, is very useful. Let q ¼ p e where p is a prime. If P1 ; P2 ; . . . ; Ps are polynomials of GFðqÞ½X1 ; X2 ; . . . ; Xn and di is the degree of Pi ði ¼ 1; 2; . . . ; sÞ, put d ¼ d1 þ d 2 þ Á Á Á þ d s and let N be the number of solutions over GFðqÞ of the system of polynomial equations: P1 ðX1 ; X2 ; . . . ; Xn Þ ¼ P2 ðX1 ; X2 ; . . . ; Xn Þ ¼ Á Á Á ¼ Ps ðX1 ; X2 ; . . .

Journal

Forum Mathematicum – de Gruyter

Published: Sep 16, 2004

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