Representation of positive semidefinite elements as sum of squares in 2-dimensional local rings

A classical problem in real geometry concerns the representation of positive semidefinite elements of a ring A as sums of squares of elements of A. If A is an excellent ring of dimension $$\ge 3$$ ≥ 3 , it is already known that it contains positive semidefinite elements that cannot be represented as sums of squares in A. The one dimensional local case has been afforded by Scheiderer (mainly when its residue field is real closed). In this work we focus on the 2-dimensional case and determine (under some mild conditions) which local excellent henselian rings A of embedding dimension 3 have the property that every positive semidefinite element of A is a sum of squares of elements of A.

Arithmetical Functions Commutable with Sums of Squares II

We give all functions ƒ , E: ℕ → ℂ which satisfy the relation for every a, b, c ∈ ℕ, where h ≥ 0 is an integers and K is a complex number. If n cannot be written as a2 + b2 + c2 + h for suitable a, b, c ∈ ℕ, then ƒ (n) is not determined. This is more complicated if we assume that ƒ and E are multiplicative functions.

Arithmetical functions commutable with sums of squares

Let k\in{\mathbb N}_0 and K\in \mathbb C, where {\mathbb N}_0, \mathbb C denote the set of nonnegative integers and complex numbers, respectively. We give all functions f, h_1, h_2, h_3, h_4:{\mathbb N}_0\to \mathbb C which satisfy the relation \[f(x_1^2+x_2^2+x_3^2+x_4^2+k)=h_1(x_1)+h_2(x_2)+h_3(x_3)+h_4(x_4)+K\] for every x_1, x_2, x_3, x_4\in{\mathbb N}_0. We also give all arithmetical functions F, H_1, H_2, H_3, H_4:{\mathbb N}\to \mathbb C which satisfy the relation \[F(x_1^2+x_2^2+x_3^2+x_4^2+k)=H_1(x_1)+H_2(x_2)+H_3(x_3)+H_4(x_4)+K\] for every x_1,x_2, x_3,x_4\in{\mathbb N}, where {\mathbb N} denotes the set of all positive integers.