×

zbMATH — the first resource for mathematics

Homotopy classes of rational functions. (Classes d’homotopie de fractions rationnelles.) (French) Zbl 1151.14016
Summary: Let \(k\) be a field of characteristic not 2 and \(n\geqslant \)1 be an integer; we show that the set of “algebraic” homotopy classes of rational functions of degree \(n\) with coefficients in \(k\) can be endowed with a graded monoid structure. Moreover, there is an isomorphism between this monoid and the monoid of orbits under the action of SL\(n(k)\) of non-degenerate symmetric bilinear forms on \(k^n\), endowed with the orthogonal sum.

MSC:
14F35 Homotopy theory and fundamental groups in algebraic geometry
14A15 Schemes and morphisms
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Bourbaki, N., Algèbre. chapitre IV : polynômes et fractions rationnelles, (1950), Hermann et Cie. Paris · Zbl 0041.36701
[2] Gel’fand, I.M.; Kapranov, M.M.; Zelevinsky, A.V., Discriminants, resultants, and multidimensional determinants, (1994), Birkhäuser Boston · Zbl 0827.14036
[3] Milnor, J.; Hussemoller, D., Symmetric bilinear forms, Ergebnisse der Mathematik und ihrer grenzgebiete, vol. 73, (1973), Springer-Verlag New York-Heidelberg
[4] F. Morel, \(\mathbf{A}^1\)-Algebraic topology over a field, preprint
[5] M. Ojanguren, The Witt group and the problem of Lüroth, Dottorato di Ricerca in Matematica, ETS Editrice, Pisa, 1990
[6] Ojanguren, M., On Karoubi’s theorem: \(W(A) = W(A [t])\), Arch. math. (basel), 43, 4, 328-331, (1984) · Zbl 0552.18003
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.