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math.NT2023
Small solutions to homogeneous and inhomogeneous cubic equations
Christian Bernert
We study the solubility of cubic equations over the integers. Assuming a necessary congruence condition, the existence of such solutions is established when the -invariant of $C…
math.NT2023
The singular series of a cubic form in many variables and a new proof of Davenport's Shrinking Lemma
Christian Bernert
We study the singular series associated to a cubic form with integer coefficients. If the number of variables is at least , we prove the absolute convergence (and hence positiv…
math.NT2023
Cubic forms over imaginary quadratic number fields and pairs of rational cubic forms
Christian Bernert, Leonhard Hochfilzer
We show that every cubic form with coefficients in an imaginary quadratic number field in at least variables represents zero non-trivially. This builds on the c…