We show that it is decidable whether or not a given Q-rational series in several noncommutative variables has a cyclic image. By definition, a series r has a cyclic image if there is a rational number q such that all nonzero coefficients of r are integer powers of q.
We show that it is decidable whether or not a given Q-rational series in several noncommutative variables has a cyclic image. By definition, a series r has a cyclic image if there is a rational number q such that all nonzero coefficients of r are integer powers of q.
Let K be a complete, algebraically closed nonarchimedean valued field, and let φ(z) ∈ K(z) have degree d ≥ 2. We study how the resultant of φ varies under changes of coordinates. For γ ∈ GL₂(K), we show that the map factors through a function on the Berkovich projective line, which is piecewise affine and convex up. The minimal resultant is achieved either at a single point in , or on a segment, and the minimal resultant locus is contained in the tree in spanned by the fixed points and poles...
The problem whether each element of a sequence satisfying a fourth order linear recurrence with integer coefficients is nonnegative, referred to as the Positivity Problem for fourth order linear recurrence sequence, is shown to be decidable.
We consider the expansion of the real field by the group of rational points of an elliptic curve over the rational numbers. We prove a completeness result, followed by a quantifier elimination result. Moreover we show that open sets definable in that structure are semialgebraic.