Abstract
We show that the Pythagoras number of rings of type \({\mathbb {R}}[x,y, \sqrt{f(x,y)}]\) is infinite, provided that the polynomial f(x, y) satisfies some mild conditions.
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References
Artin, E.: Über die Zerlegung definitier Funktionen in Quadrate. Abh. Math. Sem. Hamburg 5, 100–115 (1927)
Banecki, J.: Extensions of k-regulous functions from two-dimensional varieties, arXiv:2303.02481 [math.AG]
Banecki, J., Kowalczyk, T.: Sums of even powers of k-regulous functions. Indag. Math. 34, 477–487 (2023)
Benoist, O.: On Hilbert’s 17th problem in low degree. Algebra Number Theory 11(4), 929–959 (2017)
Benoist, O.: Sums of squares in function fields over Henselian local fields. Math. Ann. 376, 683–692 (2020)
Bochnak, J., Coste, M., Roy, M.-F.: Real Algebraic Geometry. Springer, Berlin (1998)
Cassels, J.W.S.: On the representation of rational functions as sums of squares. Acta Arith. 9, 79–82 (1964)
Cassels, J.W.S., Ellison, W.S., Pfister, A.: On sums of squares and on elliptic curves over function fields. J. Number Theory 3, 125–149 (1971)
Choi, M.-D., Lam, T.Y., Reznick, B.: Sums of squares of real polynomials. In: Proceedings of Symposia in Pure Mathematics. Am. Math. Soc. 58, 103–126 (1995)
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)
Choi, M.-D., Lam, T.Y., Reznick, B., Rosenberg, A.: Sums of squares in some integral domains. J. Algebra 65, 234–256 (1980)
Fernando, J.F.: Positive semidefinite analytic functions on real analytic surfaces. J. Geom. Anal. 31, 12375–12410 (2021)
Hoffman, D.W.: Pythagoras number of fields. J. Amer. Math. Soc. 12(3), 839–848 (1999)
Mahe, L.: Level and Pythagoras number of some geometric rings. Math. Z. 204, 615–629 (1992)
Kowalczyk, T., Miska, P.: On Waring numbers of Henselian rings. arXiv:2206.09904 [math.AC]
Motzkin, T.S.: The arithmetic-geometric inequality. In: Shishia, O. (ed.) Inequalities, pp. 205–224. Academic Press, New York (1967)
Pfister, A.: Zur Darstellung definiter Funktionen als Summe von Quadraten. Invent. Math. 4, 229–337 (1965)
Pourchet, Y.: Sur la représentation en somme de carrés des polynômes à une indéterminée sur un corps de nombres algébriques. Acta Arith. 19, 89–104 (1971)
Prestel, A., Delzell, C.: Positive Polynomials: From Hilbert’s 17th Problem to Real Algebra, Springer Monographs in Mathematics. Springer- Verlag, Berlin Heidelberg (2001)
Reznick, B.: Sums of even powers of real linear forms. Mem. Amer. Math. Soc. 96 (1992)
Scheiderer, C.: On sums of squares in local rings. J. Reine Angew. Math. 540, 205–227 (2001)
Scheiderer, C.: Extreme Points of Gram Spectrahedra of Binary Forms. Discrete Comput. Geom. 67, 1174–1190 (2022)
To, W.-K., Yeung, S.-K.: Effective isometric embeddings for certain Hermitian holomorphic line bundles. J. London Math. Soc. (2) 73, 607–624 (2006)
To, W.-K., Yeung, S.-K.: Effective Łojasiewicz inequality for arithmetically defined varieties and a geometric application to bihomogeneous polynomials Int. J. Math. 20, 915–944 (2009)
Vill, J.: Gram spectrahedra of ternary quartics. J. Symb. Comput. 116, 263–283 (2023)
Vill. J.: Integrating spectahedra, arXiv:2303.05815 [math.AG]
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Błachut, K., Kowalczyk, T. Sums of Squares on Hypersurfaces. Results Math 79, 90 (2024). https://doi.org/10.1007/s00025-023-02118-8
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DOI: https://doi.org/10.1007/s00025-023-02118-8