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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
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##### 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
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