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

Katakan x = {xi, x2,...,xn} vektor dalam ruang Zn 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 (2πif (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 | fx = 0 mod q} dengan fx 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 Zp[x, y] berdarjah lima berasaskan teknik polihedron Newton yang disekutukan dengan polinomial terbabit. Polinomial berdarjah lima yang dipertimbangkan berbentuk f(x,y) = ax5 + bx4y + cx3y2 + dx2y3 + exy4 + my5 + nx + ty + k. Kata kunci: Hasil tambah eksponen, kekardinalan, saiz p–adic, polihedron Newton Let x = {xi, x2,...,xn} be a vector in a space Zn 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 (2πif (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 | fx = 0 mod q} where fx 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 Zp[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) = ax5 + bx4y + cx3y2 + dx2y3 + exy4 + my5 + nx + ty + k. Key words: Exponential sums, cardinality, p–adic sizes, Newton polyhedron