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Taylor Polynomials of Rational Functions

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Abstract

A Taylor variety consists of all fixed order Taylor polynomials of rational functions, where the number of variables and degrees of numerators and denominators are fixed. In one variable, Taylor varieties are given by rank constraints on Hankel matrices. Inversion of the natural parametrization is known as Padé approximation. We study the dimension and defining ideals of Taylor varieties. Taylor hypersurfaces are interesting for projective geometry, since their Hessians tend to vanish. In three and more variables, there exist defective Taylor varieties whose dimension is smaller than the number of parameters. We explain this with Fröberg’s Conjecture in commutative algebra.

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Acknowledgements

Aldo Conca is supported by the project PRIN 2020355B8Y “Squarefree Gröbner degenerations, special varieties and related topics” and from GNSAGA-INdAM and was supported in part by the MIUR Excellence Department Project awarded to Dipartimento di Matematica, Università di Genova, CUP D33C23001110001. Giorgio Ottaviani is member of GNSAGA-INdAM. Simone Naldi is supported by the ANR project ANR21-CE48-0006-01 “HYPERSPACE”.

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Authors and Affiliations

  1. Dipartimento di Matematica, Dipartimento di Eccellenza 2023-2027, Università di Genova, Genova, Italy

    Aldo Conca

  2. Université de Limoges, Limoges, France

    Simone Naldi

  3. Università degli Studi di Firenze, Florence, Italy

    Giorgio Ottaviani

  4. MPI-MiS Leipzig, Leipzig, Germany

    Bernd Sturmfels

  5. UC, Berkeley, USA

    Bernd Sturmfels

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  1. Aldo Conca
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  2. Simone Naldi
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  3. Giorgio Ottaviani
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  4. Bernd Sturmfels
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Correspondence to Simone Naldi.

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Dedicated to Professor Ngo Viet Trung.

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Conca, A., Naldi, S., Ottaviani, G. et al. Taylor Polynomials of Rational Functions. Acta Math Vietnam 49, 19–37 (2024). https://doi.org/10.1007/s40306-023-00514-4

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