The Hopf condition for bilinear forms over arbitrary fields

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.

Etale homotopy and sums-of-squares formulas

“…The present paper continues the sequence [12,13] in which we prove that these results also apply to characteristic p fields. The paper [12] used mod 2 motivic cohomology, and [13] used algebraic K -theory. The paper [12] used mod 2 motivic cohomology, and [13] used algebraic K -theory.…”

Motivic connectiveK-theories and the cohomology of A(1)

“…Proof: As proved in [5,Lemma 4.7] and [28], we have that Sq 1 a D b. Since Sq 2 has degree .2;1/ and a has degree .1;1/, properties of the motivic Steenrod algebra given in [28,Lemma 9.9] imply that Sq 2 a D 0.…”

“…Proof: All three claims follow in the motivic case for the same combinatorial reasons as in the classical case. The varieties DQ n and their motivic cohomology were used in [5] to study sums-of-squares formulas. Gysin sequences are a key tool for studying cohomological invariants (including motivic cohomology and algebraic K-theory) of these varieties, but we will make computations with tools of a more homotopical flavor.…”

On lower bounds on the size of sums-of-squares formulas

“…Furthermore, Atiyah proved that if c = s+1 2 , t i must be divisible by 2 c−i for all t − r < i < c. These both provide natural lower bounds of the t required for a sums-of-squares formula with given r and s. We denote these bounds by HS(r, s) and A(r, s), respectively. In recent papers by Dugger and Isaksen [1,2], these conditions were generalized to sums of squares formulas over any field of characteristic not equal to 2.…”

“…For given r and s we wish to find lower bounds on the smallest t such that there exists a formula of the form x 2 1 + · · · + x 2 r y 2 1 + · · · + y 2 s = z 2 1 + · · · + z 2 t E-mail address: dankane@math.harvard.edu. where the z i are bilinear functions of the x's and y's.…”