On the Symbolic Manipulation for the Cardinality of Certain Degree Polynomials

The research on cardinality of polynomials was started by Mohd Atan [I] when he considered a set, V ((f) under bar ;p(alpha))= {u mod p(alpha) : (f) under bar (u) congruent to 0 mod p(alpha)}, where alpha > 0 and (f) under bar =(f(1),f(2),...,f(n)). The term (f) under bar (u)congruent to (0) under bar mod p(alpha) mod pa means that we are considering all congruence equations of modulo p(alpha) 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) under bar ;p(alpha)). The above p is a prime number and Z(p) is the ring of p-adic integers, and (x) under bar = (x(1),x(2),...,x(n)). He later let N((f) under bar ;p(alpha))= card V((f) under bar ;p(alpha)). The notation N((f) under bar ;p(alpha)) 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) under bar ;p(alpha))<= mp(1/2ord pD), where D not equal 0,alpha > ord(p)D 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.