A polynomial map F = (P,Q) ∈ ℤ[x,y]² with Jacobian $JF:=P_x{Q}_y-P_y{Q}_x\equiv 1$ has a polynomial inverse with integer coefficients if the complex plane curve P = 0 has infinitely many integer points.

We prove a sufficient condition for the Jacobian problem in the setting of real, complex and mixed polynomial mappings. This follows from the study of the bifurcation locus of a mapping subject to a new Newton non-degeneracy condition.

Let $G\to GL\left(V\right)$ be a representation of a reductive linear algebraic group $G$ on a finite-dimensional vector space $V$, defined over an algebraically closed field of characteristic zero. The categorical quotient $X=V//G$ carries a natural stratification, due to D. Luna. This paper addresses the following questions:(i) Is the Luna stratification of $X$ intrinsic? That is, does every automorphism of $V//G$ map each stratum to another stratum?(ii) Are the individual Luna strata in $X$ intrinsic? That is, does every automorphism...