Witt equivalence of rings of regular functions
Abstract
In this paper we show that the rings of regular functions on two real algebraic curves over the same real closed field are Witt equivalent (i.e. their Witt rings are isomorphic) if and only if the curves have the same number of semi-algebraically connected components. Moreover, in the second part of the paper, we prove that every strong isomorphism of Witt rings of rings of regular functions can be extended to an isomorphism of Witt rings of fields of rational functions. This extension is not unique, though.
Keywords
real algebraic curves; rings of regular functions; Witt ring of ring; Witt equivalence
References
2. Carson A.B., Marshall M.A., Decomposition of Witt rings, Canad. J. Math. 34 (1982), no. 6, 1276–1302.
3. Knebusch M., On algebraic curves over real closed fields I, Math. Z. 150 (1976), no. 1, 49–70.
4. Knebusch M., On algebraic curves over real closed fields II, Math. Z. 151 (1976), no. 2, 189–205.
5. Koprowski P., On existence of tame Harrison map, Math. Slovaca 57 (2007), no. 5, 407–414.
6. Koprowski P., Local-global principle for Witt equivalence of function fields over global fields, Colloq. Math. 91 (2002), no. 2, 293–302.
7. Koprowski P., Witt equivalence of algebraic function fields over real closed fields, Math. Zeit. 242 (2002), no. 2, 323–345.
8. Koprowski P., Integral equivalence of real algebraic function fields, Tatra Mt. Math. Publ. 32 (2005), 53–61.
9. Lam T.Y., Introduction to quadratic forms over fields, volume 67 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 2005.
10. Computer algebra system: Maxima, ver. 5.17.0, http://maxima.sf.net.
11. Milnor J., Husemoller D., Symmetric bilinear forms, Springer–Verlag, New York, 1973.
12. Perlis R., Szymiczek K., Conner P.E., Litherland R., Matching Witts with global fields, In: Recent advances in real algebraic geometry and quadratic forms (Berkeley, CA, 1990/1991; San Francisco, CA, 1991), Contemp. Math. 155 (1994), 365–387.
13. Rothkegel B., Czogała A., Singular elements and the Witt equivalence of rings of algebraic integers, Ramanujan J. 17 (2008), no. 2, 185–217.
14. Rothkegel B., Czogała A., Witt equivalence of semilocal Dedekind domains in global fields, Abh. Math. Sem. Univ. Hamburg 77 (2007), 1–24.
15. Scharlau W., Quadratic and Hermitian forms, Grundlehren der Mathematischen Wissenschaften, Vol. 270, Springer–Verlag, Berlin, 1985.
16. Szymiczek K., Matching Witts locally and globally, Math. Slovaca 41 (1991), no. 3, 315–330.
17. Szymiczek K., Witt equivalence of global fields, Comm. Algebra 19 (1991), no. 4, 1125–1149.
Instytut Matematyki, Uniwersytet Śląski w Katowicach Poland
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