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Approximate roots of quasi-ordinary polynomials

Volume 572 / 2022

Beata Gryszka Dissertationes Mathematicae 572 (2022), 1-48 MSC: Primary 13F25; Secondary 13B25. DOI: 10.4064/dm841-12-2021 Published online: 2 February 2022

Abstract

The work is devoted to approximate roots of quasi-ordinary polynomials. This research topic was initiated by Abyankar and Moh in the 1970s. They considered characteristic approximate roots of Weierstrass polynomials over the power series ring in one variable.

In 2003 González Pérez extended the Abhyankar–Moh theorem to power series ring in several variables and in 2011 Brzostowski extended the Abhyankar–Moh result to non-characteristic approximate roots, but still for power series ring in one variable.

In this work we prove a result similar to González Pérez but for quasi-ordinary Weierstrass polynomials that could be reducible (González Pérez assumed irreducibility). We also generalize the Brzostowski theorem to the power series ring in several variables and to polynomials that need not be irreducible. The main tools we use are generalized Puiseux series and monomial substitutions.

Authors

  • Beata GryszkaInstitute of Mathematics
    Pedagogical University of Kraków
    Podchorążych 2
    30-084 Kraków, Poland
    e-mail

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