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On the Cardinality of the Set of Solutions to Congruence Equation Associated with Quintic Form Image
Conference paper

On the Cardinality of the Set of Solutions to Congruence Equation Associated with Quintic Form

The exponential sum associated with f is defined aswhere the sum is taken over a complete set of residues modulo q and let x = (x1, x2, ... , xn) be a vector in the space Zn with Z ring of integers and q be a positive integer, f a polynomial in x with coefficients in Z. The value of S(f; q)has been shown to depend on the estimate of the cardinality |jV|, the number of elements contained in the setwhere fx is the partial derivative of f with respect to x = (x1, x2, ..., xn). This paper will give an explicit estimate of |V| for polynomial f(x; y) in Zp[x; y] of degree five. Earlier authors have investigated similar polynomials of lowerdegrees. The polynomial that we consider in this paper is as follows:The approach is by using p-adic Newton Polyhedron technique associated with this polynomial.
Exponents of Primitive Graphs Containing Two Disjoint Odd Cycles Image
Conference paper

Exponents of Primitive Graphs Containing Two Disjoint Odd Cycles

A connected graph G is primitive provided there exists a positive integer k such that for each pair of vertices u and v in G there is a walk of length k connecting u and v. The smallest of such positive integer k is the exponent of G. A primitive graph is said to be odd primitive graph if it has an odd exponent. It is known that if G is an odd primitive graph then G contains two disjoint odd cycles. This paper discusses exponents of a class of primitivegraphs containing of exactly two disjoint odd cycles. For such graphs we characterize the odd and even primitive graphs.
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