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Sufficient conditions for a real polynomial to be a sum of squares. (English) Zbl 1149.11018

Denote by $ℝ{\left[X\right]}_{2d}=ℝ{\left[{X}_{1},\cdots ,{X}_{n}\right]}_{2d}$ the space of real polynomials in ${X}_{1},\cdots ,{X}_{n}$ of degree$\le 2d·$ Sufficient conditions, linear in the coefficients of polynomial $f={\sum }_{\alpha }{f}_{\alpha }{X}^{\alpha }\in ℝ{\left[X\right]}_{2d},$ ($\alpha \in {ℕ}^{n},$ $|\alpha |=\sum {\alpha }_{i}\le 2d$) are provided in order that $f$ be a sum of squares (sos) of polynomials.

Write $f={f}_{0}+{\sum }_{i=1}^{n}{f}_{2d{e}_{i}}{X}_{i}^{2d}+h,$ where ${f}_{0}$ is a constant, ${e}_{i}$ the i-th standard vector, and $h$ free of the monomials ${X}_{i}^{2d}·$ Let ${\Gamma }$ be the even lattice points with $|\alpha |\le 2d·$    Theorem: If the conditions i) ${f}_{0}\ge {\sum }_{\alpha \notin {\Gamma }}|{f}_{\alpha }|-{\sum }_{\alpha \in {\Gamma }}min\left\{0,{f}_{\alpha }\right\};$ and ii) ${min}_{i=1,\cdots ,n}{f}_{2d{e}_{i}}\ge {\sum }_{\alpha \notin {\Gamma }}|{f}_{\alpha }|\frac{|\alpha |}{2d}-{\sum }_{\alpha \in {\Gamma }}min\left\{0,{f}_{\alpha }\right\}\frac{|\alpha |}{2d};$ hold true, then $f$ is sos.

To prove this and similar further results the following equivalence is noted:

$f$ of degree $\le 2d$ is sos $⇔$ ${L}_{y}\left(f\right)={\sum }_{\alpha }{f}_{\alpha }{y}_{\alpha }\ge 0$ whenever the moment matrix ${M}_{d}\left(y\right)$ defined by ${M}_{d}\left(y\right)\left(\alpha ,\beta \right)=\left({y}_{\alpha +\beta }\right)$ ($|\alpha |,|\beta |\le d,$ the ${y}_{\alpha }$ reals) is positive semidefinite.

Then ${L}_{y}\left(f\right)\ge 0$ is inferred from i, ii and elementary but technical estimates for ${L}_{y}\left({X}^{\alpha }\right),$ given ${M}_{d}\left(y\right)⪰0·$ These latter are partially in J. B. Lasserre and T. Netzer [“Sos approximation of nonnegative polynomials via simple high degree perturbations”, Math. Z. 256, 99–112 (2006; Zbl 1122.13005)]. Since the author saves space at some wrong places, the reader may have to spend some time to fill in crucial (though short) arguments.

V. Powers and T. Wörmann [“An algorithm for sums of squares of real polynomials”, J. Pure Appl. Algebra 127, 99–104 (1998; Zbl 0936.11023)] (based on ideas of Choi-Lam-Reznick) have given a necessary and sufficient criterion for a polynomial to be sum of squares and proposed Tarski quantifier elimination (q.e.) for deciding the criterion. Parrilo saw that it is in fact a semidefinite programming problem if the ${f}_{\alpha }$ are numerically given. However, q.e. would creep in again if the coefficients are treated as indeterminates and likely lead to very complicated conditions. Therefore results like the theorem above are interesting.

MSC:
 11E25 Sums of squares, etc 13J30 Real algebra 12D15 Formally real fields