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Journal Articles Journal of Pure and Applied Algebra Year : 2024

On the degree of varieties of sum of squares

Andrew Ferguson
  • Function : Author
  • PersonId : 1076496
Giorgio Ottaviani
  • Function : Author
  • PersonId : 863601
Mohab Safey El Din
Ettore Turatti
  • Function : Author
  • PersonId : 1095564

Abstract

We study the problem of how many different sum of squares decompositions a general polynomial $f$ with SOS-rank $k$ admits. We show that there is a link between the variety $\mathrm{SOS}_k(f)$ of all SOS-decompositions of $f$ and the orthogonal group $\mathrm{O}(k)$. We exploit this connection to obtain the dimension of $\mathrm{SOS}_k(f)$ and show that its degree is bounded from below by the degree of $\mathrm{O}(k)$. In particular, for $k=2$ we show that $\mathrm{SOS}_2(f)$ is isomorphic to $\mathrm{O}(2)$ and hence the degree bound becomes an equality. Moreover, we compute the dimension of the space of polynomials of SOS-rank $k$ and obtain the degree in the special case $k=2$.

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Dates and versions

hal-03698213 , version 1 (17-06-2022)

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  • HAL Id : hal-03698213 , version 1

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Andrew Ferguson, Giorgio Ottaviani, Mohab Safey El Din, Ettore Turatti. On the degree of varieties of sum of squares. Journal of Pure and Applied Algebra, In press. ⟨hal-03698213⟩
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