Characterisation of a class of equations with solutions over torsion-free groups

We study equations over torsion-free groups in terms of their “t–shape” (the occurences of the variable t in the equation). A t–shape is good if any equation with that shape has a solution. It is an outstanding conjecture [5] that all t–shapes are good. In [2] we proved the conjecture for a large class of t–shapes called amenable. In [1] Clifford and Goldstein characterised a class of good t–shapes using a transformation on t–shapes called the Magnus derivative. In this note we introduce an inverse transformation called blowing up. Amenability can be defined using blowing up; moreover the connection with differentiation gives a useful characterisation and implies that the class of amenable t–shapes is strictly larger than the class considered by Clifford and Goldstein. AMS Classification 20E34, 20E22; 20E06, 20F05