Total degree bounds for Artin \(L\)-functions and partial zeta functions. (Q1565803)

Let \(X_0\) be a closed subscheme of \(\mathbb A^n\) defined over \(\mathbb F_q\) the finite field of \(q\) elements, by vanishing of \(r\) polynomials of degree at most \(d\), let \(X:=X_0\times\mathbb F,\) where \(\mathbb F\) is a fixed algebraic closure of \(\mathbb F_q\), and let \(\Lambda(f)\) stand for the number of fixed points of an endomorphism \(f: X\to X\). Further, let \(\rho\) be a finite-dimensional \(\overline{\mathbb Q}_l\)-representation of a finite group \(G\) acting on \(X_0\), for a prime \(l\) not dividing \(q\), and let \[ v_k:=\frac{1}{|G|}\sum_{g\in G}\Lambda (gF^k)\text{Tr}\,\rho (g^{-1}). \] By a theorem of Grothendieck, the Artin \(L\)-function \[ L(X_0,\rho,t) :=\exp\left(\sum^\infty_{k=1} v_k\frac{t^k} {k}\right) \] is a rational function of \(t\). The authors prove that total number of zeros and poles of that function, counted with multiplicities, does not exceed \[ 3\cdot 2^{r+1}\cdot(3+rd)^{n+1} \cdot\dim\rho \] and obtain similar estimates for the partial zeta-functions introduced in: \textit{D. Wan}, Finite Fields Appl. 7, No. 1, 238--251 (2001; Zbl 0995.14004)].