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# On the cardinality of the set of solutions to congruence equation associated with cubic form

Aminudin, S. S. and Sapar, Siti Hasana and Mohd Atan, Kamel Ariffin (2014) On the cardinality of the set of solutions to congruence equation associated with cubic form. JP Journal of Algebra, Number Theory and Applications, 33 (1). pp. 1-23. ISSN 0972-5555

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## Abstract

Let x = (x1, x2,..., xn) be a vector in the space ℚn with ℚ field of rational numbers and q be a positive integer, f a polynomial in x with coefficient in ℚ. The exponential sum associated with f is defined as S (f;q)=∑xmodq e 2πif(x)/q, where the sum is taken over a complete set of residues modulo q. The value of S(f; q) depends on the estimate of cardinality |V|, the number of elements contained in the set V= {x mod q |f x≡0mod q}, where fx f is the partial derivative of f with respect to x. In this paper, we will discuss the cardinality of the set of solutions to congruence equation associated with a complete cubic by using Newton polyhedron technique. The polynomial is of the form f(x,y)= ax3 + bx2y + cxy2 + dy3 + 3/2ax2 + bxy + 1/2cy2 + sx + ty + k.

Item Type: Article Exponential sums; Cardinality; p-adic sizes; Newton polyhedron Faculty of Science Pushpa Publishing House 34734 Nurul Ainie Mokhtar 18 Jan 2016 14:23 18 Jan 2016 14:23

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