On the local solubility of diophantine systems
نویسنده
چکیده
Let p be a rational prime number. We refine Brauer’s elementary diagonalisation argument to show that any system of r homogeneous polynomials of degree d, with rational coefficients, possesses a non-trivial p-adic solution provided only that the number of variables in this system exceeds (rd) d 1 . This conclusion improves on earlier results of Leep and Schmidt, and of Schmidt. The methods extend to provide analogous conclusions in field extensions of Qp , and in purely imaginary extensions of Q . We also discuss lower bounds for the number of variables required to guarantee local solubility. Mathematics Subject Classifications (1991). 11D72, 11G25, 11E76, (11E95, 14G20)
منابع مشابه
Diophantine Problems in Many Variables: the Role of Additive Number Theory
We provide an account of the current state of knowledge concerning diophantine problems in many variables, paying attention in particular to the fundamental role played by additive number theory in establishing a large part of this body of knowledge. We describe recent explicit versions of the theorems of Brauer and Birch concerning the solubility of systems of forms in many variables, and esta...
متن کاملArtin’s Conjecture and Systems of Diagonal Equations
We show that Artin’s conjecture concerning p-adic solubility of Diophantine equations fails for infinitely many systems of r homogeneous diagonal equations whenever r > 2.
متن کاملInvestigating the Solubility of CO2 in the Solution of Aqueous K2CO3 Using Wilson-NRF Model
Hot potassium carbonate (PC) solution in comparison with amine solution had a decreased energy of regeneration and a high chemical solubility of . To present vapor and liquid equation (VLE) of this system and predict solubility, the ion specific non-electrolyte Wilson-NRF local composition model (isNWN) was used in this study; the framework of this model was molecular. Therefore, it was suitab...
متن کاملOn the Diophantine Equation x^6+ky^3=z^6+kw^3
Given the positive integers m,n, solving the well known symmetric Diophantine equation xm+kyn=zm+kwn, where k is a rational number, is a challenge. By computer calculations, we show that for all integers k from 1 to 500, the Diophantine equation x6+ky3=z6+kw3 has infinitely many nontrivial (y≠w) rational solutions. Clearly, the same result holds for positive integers k whose cube-free part is n...
متن کامل