2004/05/10 by Vladimir Shpilrain, Shpilrain, Vladimir, Jie-Tai Yu +1
Mathematics · #14R10 #14R15 #Algebraic Geometry (math.AG) #Commutative Algebra (math.AC) #FOS: Mathematics #math.AC #math.AG #msc:14R10 #msc:14R15
paper · pdf · doi:10.48550/arxiv.math/0405179
7 pages
arxiv created 2004/05/10 · arxiv updated 2009/12/01
Let K[x,y] be the algebra of two-variable polynomials over a field K. A polynomial p=p(x, y) is called a test polynomial (for automorphisms) if, whenever ϕ(p)=p for a mapping ϕof K[x,y], this ϕmust be an automorphism. Here we show that p ∈ C[x,y] is a test polynomial if and only if p does not belong to any proper retract of C[x,y]. This has the following corollary that may have application to the Jacobian conjecture: if a mapping ϕof C[x,y] with invertible Jacobian matrix is ``invertible on one particular polynomial", then it is an automorphism. More formally: if there is a non-constant polynomial p and an injective mapping ψof C[x,y] such that ψ(ϕ(p)) =p, then ϕis an automorphism.