Newton-Puiseux Roots of Jacobian Determinants

الأصل · EN

Let f(x,y), g(x,y) denote either a pair of holomorphic function germs, or a pair of monic polynomials in x whose coefficients are Laurent series in y. A relative polar arc is a Newton-Puiseux root, x=γ(y), of the Jacobian J=fygₓ-fₓgy. We define the tree-model, T(f,g), for the pair, using the contact orders of the Newton-Puiseux roots of f and g. We then describe how the γ's climb, and where they leave, the tree. We shall also show by two examples that the way the γ's leave the tree is not an invariant of the tree; this phenomenon is in sharp contrast to that in the one function case where the tree completely determines how the polar roots split away. Our result yield a factorisation of the Jacobian determinant.