Geodesic
Rationally isotropic quadratic spaces are locally isotropic.~III
Algebra i analiz, Tome 27 (2015) no. 6, pp. 234-241.

Voir la notice de l'article provenant de la source Math-Net.Ru

Let $R$ be a regular semilocal domain containing a field such that all the residue fields are infinite. Let $K$ be the fraction field of $R$. Let $(R^n,q\colon R^n\to R)$ be a quadratic space over $R$ such that the quadric $\{q=0\}$ is smooth over $R$. If the quadratic space $(R^n,q\colon R^n\to R)$ over $R$ is isotropic over $K$, then there is a unimodular vector $v\in R^n$ such that $q(v)=0$. If $char(R)=2$, then in the case of even $n$ our assumption on $q$ is equivalent to the fact that $q$ is a nonsingular quadratic space and in the case of odd $n>2$ our assumption on $q$ is equivalent to the fact that $q$ is a semiregular quadratic space.
Keywords: quadratic form, regular local ring, isotropic vector, Grothendieck–Serre conjecture. 
@article{AA_2015_27_6_a11,
     author = {I. Panin and K. Pimenov},
     title = {Rationally isotropic quadratic spaces are locally {isotropic.~III}},
     journal = {Algebra i analiz},
     pages = {234--241},
     publisher = {mathdoc},
     volume = {27},
     number = {6},
     year = {2015},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/AA_2015_27_6_a11/}
}
TY  - JOUR
AU  - I. Panin
AU  - K. Pimenov
TI  - Rationally isotropic quadratic spaces are locally isotropic.~III
JO  - Algebra i analiz
PY  - 2015
SP  - 234
EP  - 241
VL  - 27
IS  - 6
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/AA_2015_27_6_a11/
LA  - en
ID  - AA_2015_27_6_a11
ER  - 
%0 Journal Article
%A I. Panin
%A K. Pimenov
%T Rationally isotropic quadratic spaces are locally isotropic.~III
%J Algebra i analiz
%D 2015
%P 234-241
%V 27
%N 6
%I mathdoc
%U http://geodesic.mathdoc.fr/item/AA_2015_27_6_a11/
%G en
%F AA_2015_27_6_a11
I. Panin; K. Pimenov. Rationally isotropic quadratic spaces are locally isotropic.~III. Algebra i analiz, Tome 27 (2015) no. 6, pp. 234-241. http://geodesic.mathdoc.fr/item/AA_2015_27_6_a11/

[1] Elman R., Karpenko N., Merkurjev A., The algebraic and geometric theory of quadratic forms, Amer. Math. Soc. Publ., 56, Amer. Math. Soc., Providence, RI, 2008 | MR | Zbl

[2] Knus M. A., Quadratic and hermitian forms over rings, Grundlehren Math. Wiss., 294, Springer-Verlag, Berlin, 1991 | DOI | MR | Zbl

[3] Ojanguren M., Panin I. A., “A purity theorem for the Witt group”, Ann. Sci. Ecole Norm. Sup. (4), 32:1 (1999), 71–86 | MR | Zbl

[4] Panin I., Rehmann U., “A variant of a Theorem by Springer”, Algebra i analiz, 19:6 (2007), 117–125 www.math.uiuc.edu/K-theory/0670 | MR | Zbl

[5] Panin I., “Rationally isotropic quadratic spaces are locally isotropic”, Invent. Math., 176:2 (2009), 397–403 | DOI | MR | Zbl

[6] Panin I., Pimenov K., “Rationally isotropic quadratic spaces are locally isotropic. II”, Doc. Math., 2010, Extra Vol.: Andrei A. Suslin's sixtieth birthday, 515–523 | MR | Zbl

[7] Panin I., Stavrova A., Vavilov N., “On Grothendieck–Serre's conjecture concerning principal $G$-bundles over reductive group schemes. I”, Compos. Math., 151:3 (2015), 535–567 | DOI | MR | Zbl