Siti Hasana Sapar

A Method of Estimating the p -adic Sizes of Common Zeros of Partial Derivative Polynomials Associated with a Complete Cubic Form

Let x \(=(x_{1}, x_{2}, {\ldots}, x_{n})\) be a vector in the space Q n 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 \(S (\textit{f}; q) = \Sigma_{x mod q}e^{((2i\textit{f}(x))/q)}\), where the sum is taken over a complete set of residues modulo q. The value of \(S (\textit{f}; q)\) depends on the estimate of cardinality \(|V|\), the number of elements contained in the set \(V =\{\textit{x} mod q | \textit{f}_{\textit{x}}\equiv 0 mod q\}\) where \(\textit{f}_{\textit{x}}\) is the partial derivative of f with respect to x. To determine the cardinality of V, the p-adic sizes of common zeros of the partial derivative polynomials need to be obtained. In this paper, we estimate the p-adic sizes of common zeros of partial derivative polynomials of \(\textit{f}(x,y)\) in \(Q_{\textit{p}}[x, y]\) with a complete cubic form by using Newton polyhedron technique. The polynomial is of the form \(\textit{f}(x,y)= a\textit{x}^{3}+ b\textit{x}^{2}\textit{y} + c\textit{x}\textit{y}^{2}+d\textit{y}^{3}+ \frac{3}{2} a\textit{x}^{2}+ b\textit{x}\textit{y}+\frac{1}{2}c\textit{y}^{2}+s\textit{x}+t\textit{y}+k.\)

On the cardinality of twelfth degree polynomial

Let p be a prime and f (x, y) be a polynomial in Zp[x, y]. It is defined that the exponential sums associated with f modulo a prime pα is S(f:q)=∑e2πif(x)qfor α>1, where f (x) is in Z[x] and the sum is taken over a complete set of residues x modulo positive integer q. Previous studies has shown that estimation of S (f; pα) is depends on the cardinality of the set of solutions to congruence equation associated with the polynomial. In order to estimate the cardinality, we need to have the value of p-adic sizes of common zeros of partial derivative polynomials associated with polynomial. Hence, p-adic method and newton polyhedron technique will be applied to this approach. After that, indicator diagram will be constructed and analyzed. The cardinality will in turn be used to estimate the exponential sums of the polynomials. This paper concentrates on the cardinality of the set of solutions to congruence equation associated with polynomial in the form of f (x, y) = ax12 + bx11y + cx10y2 + sx + ty + k.

On the cardinality of the set of solutions to congruence equation associated with polynomial of degree eleven

Let p be a prime and f(x, y) be a polynomial in Zp[x, y]. For α > 1, the exponential sums associated with f modulo a prime pα is defined as S(f;q)=∑e2πif(x)q. It has been shown that the estimation of S(f; pα)depends on the cardinality of the set of solutions to the congruence equation associated with the polynomial. In order to estimate the p-adic sizes of common zeros of partial derivative polynomials associated with certain class of polynomial of degree eleven, the Newton polyhedron technique will be used. Then, the indicator diagram is constructed and analyzed. Hence, the estimation of the cardinality of the set of the solutions is determined.

An explicit estimation of p-adic sizes of common zeros of partial derivative polynomials associated with a complete quartic form

In this paper we obtain an estimation of p-adic sizes of common zeros of partial derivative polynomials associated with a complete quartic polynomial by applying the Newton polyhedron technique. Such estimates are obtained by examining indicator diagrams associated with the Newton polyhedra of partial derivatives polynomials considered and applying new conditions to improve the results of earlier researchers.

An estimation of the p-adic sizes of common zeros of partial derivative polynomials of degree six

Let x¯ = (x1,x2,...,xn) be a vector in Zn with Z ring of integers and q be a positive integer, f a polynomial in x with coefficient in Z. The exponential sum associated with f is defined as S(f;q) =  ∑ xmodqe2πif(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¯modq|f¯x¯≡0¯modq} where f¯x¯ is the partial derivatives of f with respect to x. To determine the cardinality of V, the p-adic sizes of common zeros of the partial derivative polynomials need to be obtained. In this paper, we estimate the p-adic sizes of common zeros of partial derivative polynomials of f(x,y) in Zp[x,y] with a sixth degree form by using Newton polyhedron technique. The polynomial is of the form f(x,y) = ax6+bx5y+cx4y2+sx+ty+k.

An estimation of p-adic sizes of common zeros of partial derivative polynomials associated with a quartic form

In this paper we apply Newton polyhedron technique in estimating the p-adic sizes of common zeros of partial derivative polynomial associated with a quartic polynomial. It is found that the p-adic sizes of a common zeros can be determined explicitly in terms of the p-adic orders of coefficients of dominant terms of polynomial.

On the Diophantine equation

This paper discusses an integral solution (a, b, c) of the Diophantine equations x3n+y3n = 2z2n for n ≥ 2 and it is found that the integral solution of these equation are of the form a = b = t2, c = t3 for any integers t.

A METHOD OF FINDING AN INTEGRAL SOLUTION TO x3+y3 = kz4

In this article, we proved that an integral solution (a, b, c) to the equation x3+y3 = kz4 is of the form a = rs, b = rt for any two integers s, t and c = (r3ud3)1/4 for some u with (k,r) = d where k divides a3+b3 and r is a common factor of a and b.In this article, we proved that an integral solution (a, b, c) to the equation x3+y3 = kz4 is of the form a = rs, b = rt for any two integers s, t and c = (r3ud3)1/4 for some u with (k,r) = d where k divides a3+b3 and r is a common factor of a and b.

Estimation of p –Adic Sizes of Common Zeroes of Partial Derivative Polynomials Associated with a Quintic Form

Katakan x = { x i , x 2 ,..., x n } vektor dalam ruang Z n dengan Z menandakan gelanggang integer dan q integer positif, f polinomial dalam x dengan pekali dalam Z . Hasil tambah eksponen yang disekutukan dengan f ditakrifkan sebagai S ( f ; q ) = exp (2π if ( x )/ q ) yang dinilaikan bagi semua nilai x di dalam reja lengkap modulo q . Nilai S ( f ; q ) adalah bersandar kepada penganggaran bilangan unsur | V |, yang terdapat dalam set V = { x mod q | f x = 0 mod q } dengan f x menandakan polinomial-polinomial terbitan separa f terhadap x . Untuk menentukan kekardinalan bagi V , maklumat mengenai saiz p -adic pensifar sepunya perlu diperolehi. Makalah ini membincangkan suatu kaedah penentuan saiz p -adic bagi komponen (ξ,η) pensifar sepunya pembezaan separa f ( x , y ) dalam Z p [ x , y ] berdarjah lima berasaskan teknik polihedron Newton yang disekutukan dengan polinomial terbabit. Polinomial berdarjah lima yang dipertimbangkan berbentuk f ( x , y ) = ax 5 + bx 4 y + cx 3 y 2 + dx 2 y 3 + exy 4 + my 5 + nx + ty + k . Kata kunci: Hasil tambah eksponen, kekardinalan, saiz p–adic, polihedron Newton Let x = { x i , x 2 ,..., x n } be a vector in a space Z n with Z ring of integers and let q be a positive integer, f a polynomial in x with coefficients in Z . The exponential sum associated with f is defined as S ( f ; q ) = exp (2π if ( x )/ q ) where the sum is taken over a complete set of residues modulo q . The value of S ( f;q ) has been shown to depend on the estimate of the cardinality | V |, the number of elements contained in the set V = { x mod q | f x = 0 mod q } where f x is the partial derivatives of f with respect to x . To determine the cardinality of V , the information on the p -adic sizes of common zeros of the partial derivatives polynomials need to be obtained. This paper discusses a method of determining the p -adic sizes of the components of (ξ,η) a common root of partial derivative polynomials of f ( x , y ) in Z p [ x , y ] of degree five based on the p -adic Newton polyhedron technique associated with the polynomial. The quintic polynomial is of the form f ( x , y ) = ax 5 + bx 4 y + cx 3 y 2 + dx 2 y 3 + exy 4 + my 5 + nx + ty + k . Key words: Exponential sums, cardinality, p–adic sizes, Newton polyhedron