Almost-periodic solutions for a second order abstract Cauchy problem

Summary

Let <InlineEquation><EquationSource=”tex”>M</EquationSource></InlineEquation> be a number field of degree <InlineEquation><EquationSource=”tex”>m</EquationSource></InlineEquation> with ring of integers <InlineEquation><EquationSource=”tex”>\bZ_M</EquationSource></InlineEquation>. Let <InlineEquation><EquationSource=”tex”>F\in\bZ_M[X,Y]</EquationSource></InlineEquation> be a form of degree <InlineEquation><EquationSource=”tex”>n</EquationSource></InlineEquation> such that <InlineEquation><EquationSource=”tex”>F(X,1)</EquationSource></InlineEquation> has distinct roots. Let\break <InlineEquation><EquationSource=”tex”>G\in\bZ[X,Y]</EquationSource></InlineEquation> be an arbitrary polynomial of degree <InlineEquation><EquationSource=”tex”>k</EquationSource></InlineEquation>. Assuming that <InlineEquation><EquationSource=”tex”>k\le n-2m\pl 1</EquationSource></InlineEquation> if all roots of <InlineEquation><EquationSource=”tex”>F^{(i)}(X,1)</EquationSource></InlineEquation> <InlineEquation><EquationSource=”tex”>(1\le i\le n)</EquationSource></InlineEquation> are complex and <InlineEquation><EquationSource=”tex”>k\le n-4m\pl 1</EquationSource></InlineEquation> otherwise, we provide an efficient algorithm for finding all solutions <InlineEquation><EquationSource=”tex”>X,Y\in\bZ_M</EquationSource></InlineEquation>, <InlineEquation><EquationSource=”tex”>\max\b(\overline{|X|},\overline{|Y|}\,\b)\ki C</EquationSource></InlineEquation> of the inequality <InlineEquation><EquationSource=”tex”> \overline{\b|F(X,Y)\b|\!}\,\le c \cdot \overline{\b|G(X,Y)\b|\!}\,. </EquationSource></InlineEquation> We provide numerical examples with <InlineEquation><EquationSource=”tex”>m=3</EquationSource></InlineEquation> and <InlineEquation><EquationSource=”tex”>C=10^{100}</EquationSource></InlineEquation>.