Effective formulas for the local Łojasiewicz exponent
Let \(F : (\mathbb C^n,0)\to(\mathbb C^m,0)\) be a holomorphic mapping defined in a neighborhood \(0\in\mathbb C^n\). The local Łojasiewicz exponent of \(F\), denoted by \(\mathcal L_0(F)\), is defined to be the infimum of the set of all exponents \(v\in\mathbb R\) in the Łojasiewicz inequality \(|F(z)|\geq C|z|^v\) as \(|z|\to 0\) for some constant \(C > 0\). In this paper, the authors consider the case when \(F : (\mathbb C^n , 0)\to (\mathbb C^m , 0)\) is a polynomial mapping. They give an effective algorithm computing the dimension of the germ \(V (F ) = F^{-1}(0)\) at the origin and if this dimension is zero, they compute the local Lojasiewicz exponent \(\mathcal L_0(F)\). More precisely, the main result of this paper is the following. Let \(F :(\mathbb C^n,0)\to(\mathbb C^m, 0)\) be a polynomial mapping of degree \(d = \deg F\). Denote by \(\mathbb L(m, n)\) the set of all linear mappings \(\mathbb C^m\to\mathbb C^n\). For \(q\in\{0,\dots, n\}\), \(M\in\mathbb L(n, q)\) and \(L\in\mathbb L(m+q, n)\), define \[ H_{L,M}(z) = L(F(z),M(z)) + (z_1^{d^n+1},\dots,z_n^{d^n+1}). \] Let \(\Phi_q :\mathbb L(m+q, n)\times\mathbb L(n, q)\times\mathbb L(n, 1)\times\mathbb C^n\to\mathbb L(m+q, n)\times\mathbb L(n, q)\times\mathbb L(n, 1)\times\mathbb C^n\times\mathbb C\) be a mapping given by the equation \[ \Phi_q (L, M, N, z) = (L, M, N, H_{L,M} (z), N (z)). \] The mapping \(\Phi_q\) is proper and its image is an algebraic set of pure dimension \((m + q)n + nq + 2n\). So there exists a polynomial \(P_q\in\mathbb C[L, M, N, y, t]\) of the form \[ P_q (L, M, N, y, t) = \sum_{j=0}^p P_{q,j}(L, M, N, y)t^j \] such that \(P_{q,p} \neq 0\) and the zero-set \(V (P_q)\) of \(P_q\) is the image of \(\Phi_q\). There exists \(r\) with \(0\leq r < p\) such that \(\mathrm{ord}_y P_{q,j} > 0\) for \(j = 0,\dots, r\), and \(\mathrm{ord}_y P_{q,r+1} = 0\). Set \[ \Delta'(P_q)=\min_{j=0}^r \frac{\mathrm{ord}_yP_{q,j}}{r+1-j}. \] Then \[ \dim_0 V (F) = \min\{q :\frac{1}{\Delta'(P_q)}<d^n+1\}. \] If \(F\) is finite at \(0\), then \[ \mathcal L_0(F)=\frac{1}{\Delta'(P_0)}. \]
scientific article; zbMATH DE number 5909105
scientific article