Symmetric subresultants and applications

  • Authors:

  • Philippe Saux Picart;Cyril Brunie

  • Affiliations:

  • Université de Bretagne Occidentale, Département de Mathématiques, UFR Sciences, 6 av. Victor Le Gorgeu, 29285 Brest Cédex, France;-

  • Venue:
  • Journal of Symbolic Computation
  • Year:
  • 2007

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Abstract

Schur's transforms of a polynomial are used to count its roots in the unit disk. These are generalized then by introducing the sequence of symmetric subresultants of two polynomials. Although they do have a determinantal definition, we show that they satisfy a structure theorem which allows us to compute them with a type of Euclidean division. As a consequence, a fast algorithm based on a dichotomic process and FFT is designed. We prove also that these symmetric subresultants have a deep link with Toeplitz matrices. Finally, we propose a new algorithm of inversion for such matrices. It has the same cost as those already known; however it is fraction free and consequently well adapted to computer algebra.