Estimation of a complete rational trigonometric sum. I (Q1326006)

Let \(q\) be a positive integer and let \(f= f(x)= a_ n x^ n+ \cdots+ a_ 1 x+a_ 0\) be a polynomial of degree \(n\) with integral coefficients. Furthermore, \(lc(f)= a_ n\) is the leading coefficient of \(f\); \(C(f)= \text{GCD} (a_ 0,\dots, a_ n)\) is the content of \(f\); \(\deg_ p f=\max \{k: 0\leq k\leq n,\;p\nmid a_ k\}\) is the degree of the polynomial \(\text{mod } p\); \(\delta_ m (a)=1\) when \(m\mid a\) and \(\delta_ m(a) =0\) when \(m\nmid a\). Let \(S(f,q)= \sum_{x\bmod q} e(f(x)/q)\). In the present paper, new bounds on \(| S(f,q)|\) are obtained. The author mainly proves the following theorem: Let the polynomial \(f(x)\) in \(\mathbb{Z} [x]\) of degree \(n\geq 3\) be such that \(C(f- f(0))\) is relatively prime to \(q\) and the discriminant of the derivative \(D(f')\) is different from zero. Then \[ | S(f,q)|\leq 2^{\delta_ 2 (q_ 2)/2} q^{1/2} \prod_{p\mid q_ 1} (\deg_ p f-1) \prod_{p\mid q_ 2} \deg_ p (f'/ C(f')) (lc(f')^{3-n} D(f'), q/q_ 1 q_ 2^ 2)^{1/2}, \] where \(q_ 1= \prod_{p:\text{ord}_ p q=1} p\), \(q_ 2= q/q_ 1\).