Rad HAZU, Matematičke znanosti, Vol. 26 (2022), 65-89.

MATRIX MORPHOLOGY AND COMPOSITION OF HIGHER DEGREE FORMS WITH APPLICATIONS TO DIOPHANTINE EQUATIONS

Ajai Choudhry

Abstract.   In this paper we use matrices to obtain new composition identities f(x_i)f(y_i) = f(z_i), where f(x_i) is an irreducible form, with integer coefficients, of degree n in n variables (n being 3, 4, 6 or 8), and x_i, y_i, i = 1, 2, ... , n, are independent variables while the values of z_i, i = 1, 2, ... , n, are given by bilinear forms in the variables x_i, y_i. When n = 2, 4 or 8, we also obtain new composition identities f(x_i)f(y_i)f(z_i) = f(w_i) where, as before, f(x_i) is an irreducible form, with integer coefficients, of degree n in n variables while x_i, y_i, z_i, i = 1, 2, ... , n, are independent variables and the values of w_i, i = 1, 2, ... , n, are given by trilinear forms in the variables x_i, y_i, z_i, and such that the identities cannot be derived from any identities of the type f(x_i)f(y_i) = f(z_i). 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, ... , x_n) = 1 where f(x₁, ... , x_n) is a form that admits a composition identity and n = 4, 6 or 8.

2020 Mathematics Subject Classification.   11E76, 11E16, 11C20, 11D25, 11D41.

Key words and phrases.   Composition of forms, higher degree forms, threefold composition of forms, unital commutative algebra of matrices, higher degree diophantine equations.

DOI: https://doi.org/10.21857/moxpjh1n5m

References:

1. L. E. Dickson, Homogeneous polynomials with a multiplication theorem, in Comptes Rendus du Congrès International des Mathématiciens, Strasbourg, 1920, ed. H. Villat, Toulouse (1921), pp. 215-230.

2. O. C. Hazlett, Homogeneous polynomials with a multiplication theorem, Trans. Amer. Math. Soc. 31 (1929), 223-232.
   MathSciNet     CrossRef

3. P. Lancaster and M. Tismenetsky, The Theory of Matrices, Second Edition, Academic Press, San Diego, 1985.
   MathSciNet

4. C. C. MacDuffee, On the composition of algebraic forms of higher degree, Bull. Amer. Math. Soc. 51 (1945), 198-211.
   MathSciNet     CrossRef

5. S. Pumplün, Forms of higher degree permitting composition, Beitr. Algebra Geom. 52 (2011), 265-284.
   MathSciNet     CrossRef

6. R. D. Schafer, On forms of degree n permitting composition, J. Math. Mech. 12 (1963), 777-792.
   MathSciNet

7. R. D. Schafer, Forms permitting composition, Advances in Math. 4 (1970), 111-148.
   MathSciNet     CrossRef