Simon Davis

Notes on Number Theory and Discrete Mathematics

Print ISSN 1310–5132, Online ISSN 2367–8275

Volume 23, 2017, Number 2, Pages 84—90

**Download full paper: PDF, 175 Kb**

## Details

### Authors and affiliations

Simon Davis

*Research Foundation of Southern California
8837 Villa La Jolla Drive #13595
La Jolla, CA 92039
*

### Abstract

It is proven that the validity of a conjecture on the degrees of an algebraic equation consisting of three polynomials is determined by the derivatives. The result is extended to positive polynomials satisfying a generalized Fermat equation, after setting the exponents *X*, *Y* and *Z* equal to 1, and specialization to prime factors of the product of integer values of the polynomials yields the inequality equivalent to the *abc* conjecture.

### Keywords

- Degrees
- Derivative
- Positive polynomials
- Square-free factor inequality

### AMS Classification

- 11D57
- 11T55
- 65H04

### References

- v. Sz.-Nagy, J. (1950) Über Polynome mit Lauter Reellen Nullstellen, Acta Mathematica Academiae Scientarum Hungaricae, 1, 225–228.
- Davis, S. Functional Relations for the Roots of Polynomial Equations, RFSC-05-01.
- Gauss, C. F. (1850) Theorie der Algebraischen Gleichungen, Abuandlungen von der K. Gesselschaft der Wissenschaften zu Göttingen IV, 73–102.
- Lucas, F. (1874) Propriétes géométriques des fractions rationelles, Comptes Rendus, 77, 431–433.
- Darmon, H. & Granville, A. (1995). On the Equations z
^{m}= F(x, y) and Ax^{p}+By^{q}= Cz^{r}, Bull. London Math. Soc., 27 , 513–543. - Stothers, W. W. (1981) Polynomial Identities and Hauptmodulen, Quart. J. Math. Oxford Ser. II, 32, 349–370.
- Mason, R. C. (1984) Diophantine Equations over Function Fields, Cambridge, Cambridge University Press.
- Oesterle, J. (1988) Nouvelles approches du ‘theoreme’ de Fermat, Semin. Bourbaki, 40eme

Annee, 1987/88, Exp. No. 694, Asterisque 161/162, 165–188. - Masse,r D. W. (1985) Open Problems, In: – Proc. Symp. on Analytic Number Theory, ed. W. W. L. Chen, London, Imperial College.
- Welmin W. (1903) Solutions of the Indeterminate Equation um + vn = wk, Mat. Sb., 24, 633–661.
- Tijdeman, R. (1989) Diophantine Equations and Diophantine Approximations, Proc. NATO Advanced Study Institute on Number Theory and Applications, 27 April–5 May 1988, Kluwer Acad. Publ. Dordrecht, 215–243.
- Beukers, F. (1998) The Diophantine Equation Axp+Byq = Czr, Duke Math. J., 91, 61–88.
- Lando, S. K. & Alexander, K. (2004). Graphs on Surfaces and their Applications, Encyclopedia of Mathematical Science: Lower-Dimensional Topology II, Vol. 141, Springer-Verlag, Berlin.

## Related papers

## Cite this paper

Davis, S. (2017). A conjecture on degrees of algebraic equations. Notes on Number Theory and Discrete Mathematics, 23(2), 84—90.