sums of squares
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Author(s):  
José F. Fernando

AbstractA 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.


2021 ◽  
pp. 307-322
Author(s):  
Satyabrota Kundu ◽  
Sypriyo Mazumder
Keyword(s):  

2021 ◽  
Author(s):  
Jennifer Brooks ◽  
Dusty Grundmeier ◽  
Hal Schenck

2021 ◽  
Vol 107 ◽  
pp. 67-105
Author(s):  
Elisabeth Gaar ◽  
Daniel Krenn ◽  
Susan Margulies ◽  
Angelika Wiegele

2021 ◽  
Author(s):  
Imre Kátai ◽  
Bui Minh Phong

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.


Author(s):  
Grigoriy Blekherman ◽  
Rainer Sinn ◽  
Gregory G. Smith ◽  
Mauricio Velasco

2021 ◽  
Vol 27 (3) ◽  
pp. 143-154
Author(s):  
I. Kátai ◽  
◽  
B. M. Phong ◽  

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.


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