Abstract
In this chapter we look at certificates of positivity for polynomials that are positive on basic closed semialgebraic sets in \(\mathbb R\). In contrast to the higher dimensional situation, in most cases certificates of positivity exist and there are many examples of saturated preorders. We begin with the problem of finding certificates of positivity for closed intervals in \(\mathbb R\), which is well understood; here, certificates of positivity always exist. However, the existence of certificates depends on choosing the right set of generators; in contrast to Schmüdgen’s Positivstellensatz, results here are not true for any possible set of generators. Finally, we take a brief look at positivity on curves in the plane.
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References
Hermite, C.: problem. Intermèd de mat. 1, 65–66 (1894)
Bernstein, S.: Sur la représentation des polynômes positif. Soobshch. Har’k. Mat. Obshch. 2, 227–228 (1915)
Powers, V., Reznick, B.: Polynomials that are positive on an interval. Trans. Amer. Math. Soc. 352, 4677–4692 (2000)
Pólya, G., Szegő, G.: Problems and Theorems in Analysis II. Classics in Mathematics. Springer, Berlin (1998); Theory of functions, zeros, polynomials, determinants, number theory, geometry, Translated from the German by C. E. Billigheimer, Reprint of the 1976 English translation
Stengle, G.: Complexity estimates for the Schmüdgen Positivstellensatz. J. Complexity 12, 167–174 (1996)
Algorithms in Real Algebraic Geometry, 2nd edn. Springer, Berlin (2006). Revised version of the first edition online at https://perso.univ-rennes1.fr/marie-francoise.roy/bpr-ed2-posted3.pdf
Boudaoud, F., Caruso, F., Roy, M.-F.: Certificates of positivity in the Bernstein basis. Discrete Comput. Geom. 39, 639–655 (2008)
Kuhlmann, S., Marshall, M.: Positivity, sums of squares and the multidimensional moment problem. Trans. AMS 354, 4285–4301 (2002)
Berg, C., Maserick, P.H.: Polynomially positive definite sequences. Math. Ann. 259, 487–495 (1982)
Scheiderer, C.: Distinguished representations of non-negative polynomials. J. Algebra 289, 558–573 (2005)
Marshall, M.: Positive Polynomials and Sums of Squares. Mathematical Surveys and Monographs, vol. 146. American Mathematical Society, Providence (2008)
Choi, M.D., Dai, Z.D., Lam, T.Y., Reznick, B.: The Pythagoras number of some affine algebras and local algebras. J. Reine Angew. Math. 336, 45–82 (1982)
Fejér, L.: Über trigonometrische Polynome. J. Reine Angew. Math. 146, 53–82 (1916)
Scheiderer, C.: Sums of squares of regular function on real algebraic varieties. Trans. Amer. Math. Soc. 352, 1029–1069 (2000)
Scheiderer, C.: Sums of squares on real algebraic curves. Math. Z. 245, 725–760 (2003)
Plaumann, D.: Sums of squares on reducible real curves. Math. Z. 265, 777–797 (2010)
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Powers, V. (2021). The Dimension One Case. In: Certificates of Positivity for Real Polynomials. Developments in Mathematics, vol 69. Springer, Cham. https://doi.org/10.1007/978-3-030-85547-5_8
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DOI: https://doi.org/10.1007/978-3-030-85547-5_8
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