Sums of squares in function fields over Henselian local fields

Automorphic Forms and Sums of Squares over Function Fields

We develop some of the theory of automorphic forms in the function field setting. As an application, we find formulas for the number of ways a polynomial over a finite field can be written as a sum of k squares, k ≥ 2. Given a finite field Fq with q odd, we want to determine how many ways a polynomial in Fq[T ] can be written as a sum of k squares. For k ≥ 3 (or k = 2, −1 not a square in Fq), t...

Henselian Function Fields

Sums of squares in excellent henselian local rings

This is a survey about the representation of positive semidefinite elements as sums of squares from a local viewpoint, in the sense of analytic geometry. Here, we present the most recent results in the more general frame of excellent henselian local rings with real closed residue field. 1

Sums of Squares in Function Fields of Quadrics and Conics

For a quadric Q over a real field k, we investigate whether finiteness of the Pythagoras number of the function field k(Q) implies the existence of a uniform bound on the Pythagoras numbers of all finite extensions of k. We give a positive answer if the quadratic form that defines Q is weakly isotropic. In the case where Q is a conic, we show that the Pythagoras number of k(Q) is 2 only if k is...

Bounding the Rational Sums of Squares over Totally Real Fields

Let K be a totally real Galois number field. C. J. Hillar proved that if f ∈ Q[x1, . . . , xn] is a sum of m squares in K[x1, . . . , xn], then f is a sum of N(m) squares in Q[x1, . . . , xn], where N(m) ≤ 2[K:Q]+1 · `[K:Q]+1 2 ́ ·4m, the proof being constructive. We show in fact that N(m) ≤ (4[K : Q]−3)·m, the proof being constructive as well.