Estimation of the number of the critical values at infinity of a polynomial function \(f:\mathbf C^ 2 \to \mathbf C\)
Let \(f : \mathbb{C}^2 \to \mathbb{C}\) be a polynomial map. \(t \in \mathbb{C}\) is called a regular value at infinity of \(f\), if there exist positive numbers \(R\) and \(\varepsilon\) such that the restriction of \(f\), \(f : f^{-1}(D_\varepsilon (t)) \setminus B(R) \to D_\varepsilon (t)\), is a trivial fibration, \(D_\varepsilon (t) = \{z \in \mathbb{C} \mid |z - t|\leq \varepsilon\}\), \(B(R) = \{(x,y) \in \mathbb{C}^2 \mid |x|^2 + |y |^2 \leq R\}\). Otherwise \(t\) is called a critical value at infinity. The number of the points of the set of critical values at infinity of \(f\) is estimated. Let \(f = \sum^d_{v = 0} f_v\), \(f_v\) homogeneous of degree \(v\) and \(f_d = cx^{v_0} y^{v_{k+1}} \prod^k_{j = 1} (y - \lambda_j x)^{v_j}\), \(\lambda_i, c \in \mathbb{C}^*\) and the \(\lambda_i\) distinct. Then \(\sum^{k + 1}_{i = 0} \max \{v_i - 1,0\}\) (the projective degeneracy at infinity) is a bound for the number of critical values at infinity. This estimation is sharpened by the so-called toric degeneracy which uses more information than \(f_d\). Toric compactifications and affine polar invariants are used for the proof.
scientific article; zbMATH DE number 878490