TAILIEUCHUNG - Đề tài " The Hopf condition for bilinear forms over arbitrary fields "

We settle an old question about the existence of certain ‘sums-of-squares’ formulas over a field F , related to the composition problem for quadratic forms. A classical theorem says that if such a formula exists over a field of characteristic 0, then certain binomial coefficients must vanish. We prove that this result also holds over fields of characteristic p | Annals of Mathematics The Hopf condition for bilinear forms over arbitrary fields By Daniel Dugger and Daniel C. Isaksen Annals of Mathematics 165 2007 943 964 The Hopf condition for bilinear forms over arbitrary fields By Daniel Dugger and Daniel C. Isaksen Abstract We settle an old question about the existence of certain sums-of-squares formulas over a field F related to the composition problem for quadratic forms. A classical theorem says that if such a formula exists over a field of characteristic 0 then certain binomial coefficients must vanish. We prove that this result also holds over fields of characteristic p 2. 1. Introduction Fix a field F. A classical problem asks for what values of r s and n do there exist identities of the form r C1 1 I 2 - É y2 Y z i 1 i 1 i 1 where the Zi s are bilinear expressions in the x s and y s. Equation is to be interpreted as a formula in the polynomial ring F x1 . xr y1 . ys we call it a sums-of-squares formula of type r s n . The question of when such formulas exist has been extensively studied L and S1 are excellent survey articles and S2 is a detailed sourcebook. In this paper we prove the following result solving Problem C of L Theorem . If F is afield of characteristic not equal to 2 and a sums-of-squares formula of type r s n exists over F then n must be even for n r i s. We now give a little history. It is common to let r F s denote the smallest n for which a sums-of-squares formula of type r s n exists. Many papers have studied lower bounds on r F s but for a long time such results were known only for fields of characteristic 0 one reduces to a geometric problem over R and then topological methods are used to obtain the bounds see L for a summary . In this paper we begin the process of extending such results to characteristic p replacing the topological methods by those of motivic homotopy theory. 944 DANIEL DUGGER AND DANIEL C. ISAKSEN The most classical result along these lines is Theorem for the .

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