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We prove two results related to the solutions of norm form equations. First, we give a finiteness result for sums of terms of linear recurrence sequences appearing in the coordinates of solutions of norm form equations. Second, we give a finiteness result concerning solutions of norm form equations representable as sums of S-units with a fixed number of terms. To prove these results, we use a deep result concerning the finiteness of the solutions of polynomial-exponential equations and S-unit equations.

A multivariate, formal power series over a field K is a Bézivin series if all of its coefficients can be expressed as a sum of at most r elements from a finitely generated subgroup $G \le K^*$; it is a Pólya series if one can take $r=1$. We give explicit structural descriptions of D-finite Bézivin series and D-finite Pólya series over fields of characteristic $0$, thus extending classical results of Pólya and Bézivin to the multivariate setting.

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SUMS OF TERMS OF RECURRENCE SEQUENCES AND S-UNITS IN THE SOLUTION SETS OF NORM FORM EQUATIONS

D-finite multivariate series with arithmetic restrictions on their coefficients

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2 results

SUMS OF TERMS OF RECURRENCE SEQUENCES AND S-UNITS IN THE SOLUTION SETS OF NORM FORM EQUATIONS

D-finite multivariate series with arithmetic restrictions on their coefficients