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Let $\mathbb{R}[X]_{\leq 2d}$ denote the real vector space of polynomials of degree at most $2d$ in the coordinate ring $\mathbb{R}[X]$ of variety $X$.

Definition: A polynomial $f$ is $d$-SOS if there exist $g_{1}, \dots, g_{k} \in \mathbb{R}[X]_{d}$ such that $$f = g^{2}_{1} + \cdots + g^{2}_{k}$$

Question: Suppose that you have a finite set $X\subset \mathbb{R}^{n}$ and a sequence $\{f_{i}\}_{i\in \mathbb{N}}$ of $d$-SOS polynomials in $\mathbb{R}[X]_{\leq 2d}$, then is $$f := \lim_{i \in \mathbb{N}} f_{i}$$ also $d$-SOS?

For me, it's clear that any non-negative polynomial in $X$, as it is finite, is sum of squares (SOS), but for me it is not clear why the limit has degree at most $2d$. Is it true? Why?

Rodrigo de Azevedo
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Tio Miserias
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  • Tio, please remind me, what it means to be a polynomial in the coordinate ring of a finite set $X$. How does it differ from simply being a polynomial? – Gerry Myerson Jul 24 '20 at 02:50
  • $\mathfrak{I}(X)={f\in \mathbb{R}[x_{1},\dots, x_{n}]: \forall p\in X, f(p)=0 }$. The coordinate ring of $X$ is $\mathbb{R}[x_{1},\dots, x_{n}]/ \mathfrak{I}(X)$ and it is denoted by $\mathbb{R}[X]$ – Tio Miserias Jul 24 '20 at 02:59
  • OK, so, the elements of the coordinate ring are not polynomials, but cosets of an ideal whose elements are polynomials. So any one element of the coordinate ring will contain polynomials of infinitely many different degrees. So now I'm not sure how one assigns a degree to an element of the coordinate ring. – Gerry Myerson Jul 24 '20 at 03:16
  • In any event, it might be worth your while to look at the Motzkin example at the link I gave in a comment to the answer from JCAA, and follow the links at that link, to see whether you can make any use of that line of enquiry. Since JCAA has deleted the answer, I'll repeat the link here: https://math.stackexchange.com/questions/1162920/why-multivaraite-positive-polynomials-cannot-be-written-as-sum-of-squares? – Gerry Myerson Jul 24 '20 at 03:22
  • The degree of the coset is the degree of the polynomial with minimal degree in the coset. And the main representative of this coset will be this polynomial. – Tio Miserias Jul 24 '20 at 03:23
  • The Motzkin example is useful when $X=\mathbb{R}^{3}$, but here $X$ is any finite set, and it is a little bit different. – Tio Miserias Jul 24 '20 at 03:25
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    Hint: a) values of polynomials of fixed degree on $X$ form a linear space; b) If you have a square of a fixed linear form in variables $x_j$, it is just a fixed linear form in $u_{ij}=x_ix_j$; c) linear combinations of various $u_{ij}$ of this type with non-negative coefficients are just all all non-negative definite matrices. – fedja Jul 29 '20 at 21:18
  • @TioMiserias Do you agree with my edits? – Rodrigo de Azevedo Aug 31 '20 at 13:26

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