Relationship between polynomials with multiple roots and rational functions with common roots

• Autores: Y. Kamiyama
• Localización: Mathematica scandinavica, ISSN 0025-5521, Vol. 96, Nº 1, 2005, págs. 31-48
• Idioma: inglés
• DOI: 10.7146/math.scand.a-14943
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• Resumen

  • or F=R or C, let \Plk,n(F)l denote the space of monic polynomials f(z) over F of degree k and such that the number of n-fold roots of f(z) is at most l. Let Xlk,n(F) denote the space consisting of all n-tuples (p1(z),…,pn(z)) of monic polynomials over F of degree k and such that there are at most l roots common to all pi(z). In this paper, we prove that Plk,n(F) and Xl[kn],n(F) are stably homotopy equivalent. In fact, they are homotopy equivalent when F=C and (n,l)≠(2,0). We also consider the case that n-fold roots and common roots are not real. These results generalize previous results concerning these spaces.