Estimates for nonsingular multiplicative character sums (Q2782259)

Let \(\mathbb{F}_{q}\) be a finite field of characteristic \(p\) with \(q\) elements, \( \chi\) a nontrivial multiplicative character of \(\mathbb{F}_{q}\), and \(f(x)=f(x_{1}, \dots ,x_{n})\) a polynomial in \(\mathbb{F}_q[x_{1}, \dots ,x_{n}]\) of degree \(d \geq 1\), say NEWLINE\[NEWLINE f=f_{d}+f_{d-1}+ \cdots+f_{0} NEWLINE\]NEWLINE with \(f_{i}\) homogeneous of degree \(i\). The author studies character sums NEWLINE\[NEWLINE S(f, \chi)= \sum_{x \in\mathbb{F}_{q}^{n}} \chi((f(x)) NEWLINE\]NEWLINE and associated \(L\)-functions \(L(f, \chi,T)\) of a complex variable \(T\), for various \(f\) and \( \chi\). NEWLINENEWLINENEWLINEIf \((d,p)=1\) and the equation \(f_{d}=0\) defines a smooth projective hypersurface in \({\mathbb P}^{n-1}\) of degree \(d\), then Deligne's result implies that \(L(f, \chi,T)\) is a polynomial \(P_{n}(T)\) (for \(n\) odd) or a reciprocal polynomial \(1/P_{n}(T)\) (for \(n\) even) of degree \((d-1)^{n}\); moreover, the reciprocal roots \( \theta_{i,n}\) of \(P_{n}(T)\) are algebraic numbers with the same absolute value \( \sqrt{q^{n}}\). Since NEWLINE\[NEWLINE S(f)=(-1)^{n} \sum_{i=1}^{(d-1)^{n}} \theta_{i,n} NEWLINE\]NEWLINE this gives the estimate NEWLINE\[NEWLINE |S(f)|\leq (d-1)^{n} \sqrt{q^{n}}. NEWLINE\]NEWLINE NEWLINENEWLINENEWLINESuppose that NEWLINENEWLINENEWLINE\qquad \((a)\) the equation \(f_{d}=0\) defines a smooth projective hypersurface in \({\mathbb P}_{\mathbb{F}_{q}}^{n-1}\) NEWLINENEWLINENEWLINE\qquad \qquad of degree \(d\), NEWLINENEWLINENEWLINE\qquad \((b)\) the equation \(f=0\) defines a smooth hypersurface in \( {\mathbb A}_{\mathbb{F}_{q}}^{n}\). NEWLINENEWLINENEWLINEIn this situation the author refines the above result of Deligne and proves that if \((d,p)=1\) and \( \chi^{d}\) is nontrivial, then NEWLINE\[NEWLINE |S(f, \chi)|\leq (d-1)^{n} \sqrt{q^{n}}, NEWLINE\]NEWLINE the associated \(L\)-function \(L(f, \chi,T)\) is a polynomial \(P_{n}(T)\) (for \(n\) odd) or a reciprocal polynomial \(1/P_{n}(T)\) (for \(n\) even) of degree \((d-1)^{n}\), and reciprocal roots of \(P_{n}(T)\) have the same absolute value \( \sqrt{q^{n}}\). On the other hand, if \( \chi^{d}\) is trivial, then NEWLINE\[NEWLINE |S(f, \chi)|\leq (d-1)^{n} \sqrt{q^{n}}, NEWLINE\]NEWLINE the associated \(L\)-function \(L(f, \chi,T)\) is a polynomial \(P_{n}(T)\) (for \(n\) odd) or a reciprocal polynomial \(1/P_{n}(T)\) (for \(n\) even) of degree \((d-1)^{n}\), and \(P_{n}(T)\) has \(((d-1)^{n+1}-(-1)^{n+1})/d\) reciprocal roots with absolute value \( \sqrt{q^{n}}\) and \(((d-1)^{n}-(-1)^{n})/d\) reciprocal roots with absolute value \( \sqrt{q^{n-1}}\). NEWLINENEWLINENEWLINEThe author gives also several generalizations of the above result in terms of smooth projective and geometrically connected \(\mathbb{F}_{q}\)-schemes.