#### Rad Hrvatske akademije znanosti i umjetnosti. Matematičke znanosti, No. 551=26, 2022.

Izvorni znanstveni članak

https://doi.org/10.21857/moxpjh1n5m

Matrix morphology and composition of higher degree forms with applications to diophantine equations

Ajai Choudhry ; 13/4 A Clay Square, Lucknow - 226001, India

Puni tekst:

str. 65-89

preuzimanja: 205

###### Sažetak

In this paper we use matrices to obtain new composition identities f(xi)f(yi) = f(zi), where f(xi) is an irreducible form, with integer coefficients, of degree n in n variables (n being 3, 4, 6 or 8), and xi, yi, i = 1, 2, ... , n, are independent variables while the values of zi, i = 1, 2, ... , n, are given by bilinear forms in the variables xi, yi. When n = 2, 4 or 8, we also obtain new composition identities f(xi)f(yi)f(zi) = f(wi) where, as before, f(xi) is an irreducible form, with integer coefficients, of degree n in n variables while xi, yi, zi, i = 1, 2, ... , n, are independent variables and the values of wi, i = 1, 2, ... , n, are given by trilinear forms in the variables xi, yi, zi, and such that the identities cannot be derived from any identities of the type f(xi)f(yi) = f(zi). Further, we describe a method of obtaining both these types of composition identities for forms of higher degrees. We also describe a method of generating infinitely many integer solutions of certain quartic and octic diophantine equations f(x1, ... , xn) = 1 where f(x1, ... , xn) is a form that admits a composition identity and n = 4, 6 or 8.

283928

27.9.2022.

Posjeta: 389 *