Abstract
It is not difficult to prove that any polynomial p(x) with real coefficients which takes non-negative values for all x ε ℝ can be represented as the sum of squares of two polynomials with real coefficients. Indeed, the roots of a polynomial with real coefficients can be divided into the real roots and pairs of complex conjugate ones. Therefore
where \(\alpha_k \in \mathbb{R}. \; {\rm If} \; p(x)\geq 0 \; {\rm for \ all}\; x \in \mathbb{R}, \; {\rm then} \; a \geq 0\) and all the numbers m k are even, and so the real roots also split into pairs. Hence
where some of the z j can be real. Let
where q and r are polynomials with real coefficients. Then
As a result, we obtain \(p(x)=(q(x))^2 + (r(x))^2.\)
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2004 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Prasolov, V.V. (2004). Hilbert’s Seventeenth Problem. In: Polynomials. Algorithms and Computation in Mathematics, vol 11. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-03980-5_7
Download citation
DOI: https://doi.org/10.1007/978-3-642-03980-5_7
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-03979-9
Online ISBN: 978-3-642-03980-5
eBook Packages: Springer Book Archive