Waring's problem for the ring of polynomials

Newman, Donald J.; Slater, M

doi:10.1016/0022-314x(79)90027-1

Cited by 31 publications

(24 citation statements)

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“…Most of our results deal with the secant degeneracy index μ . However, the second part of Theorem 1 provides a non-trivial lower bound for μ generalizing a similar result of a well-known paper Newman and Slater (1979) from 1979 where the special case of partitions with equal parts was considered.…”

Section: Problem 1 For a Given Partition μ D Calculate/estimate Its mentioning

confidence: 56%

Secant Degeneracy Index of the Standard Strata in The Space of Binary Forms

Nenashev

Shapiro

2017

Arnold Math J.

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The space Pol dCP d of all complex-valued binary forms of degree d (considered up to a constant factor) has a standard stratification, each stratum of which contains all forms whose set of multiplicities of their distinct roots is given by a fixed partition μ d. For each such stratum S μ , we introduce its secant degeneracy index μ which is the minimal number of projectively dependent pairwise distinct points on S μ , i.e., points whose projective span has dimension smaller than μ − 1. In what follows, we discuss the secant degeneracy index μ and the secant degeneracy index μ of the closureS μ .

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Section: Problem 1 For a Given Partition μ D Calculate/estimate Its mentioning

confidence: 56%

Secant Degeneracy Index of the Standard Strata in The Space of Binary Forms

Nenashev

Shapiro

2017

Arnold Math J.

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“…• [1,5,6,17,18,21]. Next we state the one dimensional version of two of these extensions, proved in [6] for polynomials in several variables, that we will use in the proof of Theorem B.…”

Section: Generalized Fermat Type Theorems For Polynomialsmentioning

confidence: 99%

“…When n = 1, P (u, v) has no singular points and hence P (u, v) is irreducible. Using again formula (17) we get that…”

Section: Proposition 42 Consider the Polynomialmentioning

confidence: 99%

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On the number of polynomial solutions of Bernoulli and Abel polynomial differential equations

Cima

Gasull

Mañosas

2017

Journal of Differential Equations

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Abstract. In this paper we determine the maximum number of polynomial solutions of Bernoulli differential equations and of some integrable polynomial Abel differential equations. As far as we know, the tools used to prove our results have not been utilized before for studying this type of questions. We show that the addressed problems can be reduced to know the number of polynomial solutions of a related polynomial equation of arbitrary degree. Then we approach to these equations either applying several tools developed to study extended Fermat problems for polynomial equations, or reducing the question to the computation of the genus of some associated planar algebraic curves.

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“…Many scholars have investigated this and related problems, for details please see [3,4,12,14,15,18,20,23,25] etc. Gundersen-Hayman [6] collected the best lower 908 Nan Li estimates that are known for every n as follows:…”

Section: Introductionmentioning

confidence: 99%