On zeros of a system of quadratic forms in 3 variables

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• 作者：A.S. Sivatski ^{alexander.sivatski@gmail.com" class="auth_mail" title="E-mail the corresponding author}
• 关键词：11E04 ; 11F23
• 刊名：Journal of Algebra
• 出版年：2016
• 出版时间：1 March 2016
• 年：2016
• 卷：449
• 期：Complete
• 页码：237-245
• 全文大小：280 K

文摘

Let k   be a field of characteristic distinct from 2, f₁$f_1$, f₂$f_2$, f₃$f_3$ quadratic forms in 3 variables over k. We prove that these forms have a common zero over k   if and only if the quadratic form u₁f₁+u₂f₂+u₃f₃$u_1{f}_1+u_2{f}_2+u_3{f}_3$ is isotropic over the field k(u₁,u₂,u₃)$k(u_1,u_2,u_3)$ and the cubic form det(t₁f₁+t₂f₂+t₃f₃)$\det (t_1{f}_1+t_2{f}_2+t_3{f}_3)$ in variables t₁$t_1$, t₂$t_2$, t₃$t_3$ is isotropic over k. We also consider a similar question for n   quadratic forms in 3 variables where n≥4$n\ge 4$.