On the resolution of index form equations in biquadratic number fields. II
[For part I, cf. the preceding review.] The authors present a method to solve completely index form equations (1) \(I(x_ 2,x_ 3,x_ 4)=\pm 1\), where I is the index form of some integral basis \(\{1,w_ 2,w_ 3,w_ 4\}\) of some totally real cyclic biquadratic number field K. I is the product of linear forms with algebraic coefficients. Between these linear forms there are several linear relations of the type \(c_ 1L_ 1+c_ 2L_ 2+c_ 3L_ 3=0\), where \(L_ 1,L_ 2,L_ 3\) are linear factors of I. If \((x_ 2,x_ 3,x_ 4)\) is a solution in integers of (1), then each \(L_ i(x_ 2,x_ 3,x_ 4)\) is essentially an algebraic unit. Hence (1) can be reduced to a number of equations \(c_ 1\epsilon_ 1+c_ 2\epsilon_ 2+c_ 3\epsilon_ 3=0\) in units \(\epsilon_ 1,\epsilon_ 2,\epsilon_ 3\). By some standard argument, one can reduce such a unit equation to an inequality involving a linear form in logarithms of algebraic numbers with integer coefficients. Then Baker's method yields an effective upper bound for the sizes of the coefficients of such a linear form in logarithms. The authors give a method, based on some variation of the Baker-Davenport lemma and the LLL-lattice basis reduction algorithm, which in many cases enables one to reduce the bound arising from Baker's method to a more tractable size and then to find all solutions of (1). The authors give some numerical examples.
linear form in logarithms of algebraic numbers
scientific article; zbMATH DE number 4198082
scientific article