On the symbolic manipulation for the cardinality of certain degree polynomials

Abstract

The research on cardinality of polynomials was started by Mohd Atan [1] when he considered a set, V(f_;pα)={umodpα:f_(u)≅0modpα}, where α > 0 and f_=(f1,f2,⋯fn). The term f_(u)≅0_modpα means that we are considering all congruence equations of modulo pα and we are looking for u that makes the congruence equation equals zero. This is called the zeros of polynomials. The total numbers of such zeros is termed as N(f_;pα). The above p is a prime number and Zp is the ring of p-adic integers, and x_=(x1,x2, ⋯xn). He later let N(f_;pα)=card V(f_;pα). The notation N(f_;pα) means the number of zeros for that the polynomials f_. For a polynomial f (x) defined over the ring of integers Z, Sandor [2] showed that N(f;pα)≤ mp12ordpD, where D ≠ 0, α > ordpD and D is the discriminant of f. In this paper we will try to introduce the concept of symbolic manipulation to ease the process of transformation from two-variables polynomials to one-variable polynomials.