On the Diophantine equation \(y^p = f(x_1, x_2,\ldots, x_r)\). (English) Zbl 1435.11065

Srikanth, Raghavendran; Subburam, Sivanarayanapandian

Funct. Approximatio, Comment. Math. 58, No. 1, 37-42 (2018).

Let \(R\) be a subgroup of the additive group of real numbers having the least element property, \(p\) a prime number and \(f(x_1,\ldots,x_t)\) a polynomial of the form \(f(x_1,\ldots,x_r) = B(x_1,\ldots, x_t)^p+C(x_1,\ldots, x_t)\), where \(B(x_1,\ldots, x_t)\) and \(C(x_1,\ldots, x_t)\) are polynomials with real coefficients. In this paper the solutions of the equation \(y^p = f(x_1, x_2,\ldots,x_r)\) over \(R\) are studied and an estimate for their size is given.

Reviewer: Dimitros Poulakis (Thessaloniki)

MSC:

11D41	Higher degree equations; Fermat’s equation
11D72	Diophantine equations in many variables

Keywords:

higher degree Diophantine equation

References:

[1]	D. Poulakis, A simple method for solving the Diophantine equation \(y^{2} = x^{4} + ax^{3} + bx^{2} + cx + d\), Elem. Math. 54 (1999), 32-36. · Zbl 0952.11005
[2]	A. Sankaranarayanan and N. Saradha, Estimates for the solutions of certain Diophantine equations by Runge’s method, Int. J. of Number Theory 4(3) (2008), 475-493. · Zbl 1165.11035
[3]	R. Srikanth and S. Subburam, The superelliptic equation \(y^{p} = x^{kp} + a_{kp - 1}x^{kp - 1} + ⋯ + a_{0}\), Journal of Algebra and Number Theory Academia 2(6) (2012), 331-385.
[4]	S. Subburam, The Diophantine equation \((y-q_1) (y-q_2)⋯(y-q_m) = f (x)\), Acta Math. Hungar. 146(1) (2015), 40-46. · Zbl 1374.11052
[5]	S. Subburam and R. Thangadurai, On the Diophantine equation \(x^3 + by + 1 - xyz = 0\), C. R. Math. Rep. Acad. Sci. Canada 36(1) (2014), 15-19. · Zbl 1361.11019
[6]	S. Subburam and R. Thangadurai, On the Diophantine equation \(ax^3 + by + c = xyz\), Funct. Approx. Comment. Math. 53(1), (2015), 167-175. · Zbl 1397.11072
[7]	L. Szalay, Fast algorithm for solving superelliptic equations of certain types, Acta Acad. Paedagog. Agriensis Sect. Mat. (N.S.) 27 (2000), 19-24 · Zbl 0973.11039
[8]	L. Szalay, Superelliptic equations of the form \(y^{p} = x^{kp} + a_{kp - 1}x^{kp - 1} +⋯+a_{0}\), Bull. Greek Math. Soc. 46 (2002), 23-33. · Zbl 1014.11020
[9]	L. Szalay, Algorithm to solve ternary Diophantine equations, Turkish Journal of Mathematics 37 (2013), 733-738. · Zbl 1362.11042