“Mason's Theorem”的概念、定义、习题解题方法及翻译-数学辞典

单词

Mason's Theorem

释义

Mason's Theorem

Let there be three Polynomials , , and with no common factors such that

Then the number of distinct Roots of the three Polynomials is one or more greaterthan their largest degree. The theorem was first proved by Stothers (1981).

Mason's theorem may be viewed as a very special case of a Wronskian estimate (Chudnovsky and Chudnovsky 1984). Thecorresponding Wronskian identity in the proof by Lang (1993) is

so if

, and

are linearly dependent, then so are

and

. More powerful Wronskian estimates withapplications toward diophantine approximation of solutions of linear differential equations may be found in Chudnovskyand Chudnovsky (1984) and Osgood (1985).

The rational function case of Fermat's Last Theorem follows trivially from Mason's theorem (Lang 1993, p. 195).

See also abc Conjecture

References

Chudnovsky, D. V. and Chudnovsky, G. V. ``The Wronskian Formalism for Linear Differential Equations and Padé Approximations.'' Adv. Math. 53, 28-54, 1984.

Lang, S. ``Old and New Conjectured Diophantine Inequalities.'' Bull. Amer. Math. Soc. 23, 37-75, 1990.

Lang, S. Algebra, 3rd ed. Reading, MA: Addison-Wesley, 1993.

Osgood, C. F. ``Sometimes Effective Thue-Siegel-Roth-Schmidt-Nevanlinna Bounds, or Better.'' J. Number Th. 21, 347-389, 1985.

Stothers, W. W. ``Polynomial Identities and Hauptmodulen.'' Quart. J. Math. Oxford Ser. II 32, 349-370, 1981.

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