##
**Equations over finite fields. An elementary approach.**
*(English)*
Zbl 0329.12001

Lecture Notes in Mathematics. 536. Berlin-Heidelberg-New York: Springer-Verlag. ix, 267 p. (1976).

The Lecture Notes consist of introduction: Chapter I: Equations \(y^4=f(x)\) and \(y^q-y= f(x)\); Chap. II: Character Sums and Exponential Sums; Chap. III: Absolutely Irreducible Equations \(f(x,y)=0\); Chap. IV: Equations in Many Variables; Chap. V: Absolutely Irreducible Equations \(f(x_1,\dots,x_n)= 0\); Chap. VI: Rudiments of Algebraic Geometry. The Number of Points in Varieties over Finite Fields; and Bibliography.

Let \(f(x)\) be a polynomial in \(x\) with coefficients in a finite field \(\mathbb F_q\) of \(q\) elements. E. Artin (1924) conjectured and H. Hasse (1936) proved that the number \(N\) of solutions \((x,y)\in \mathbb F_q\times \mathbb F_q\) of the equation \(y^2= f(x)\) satisfies \[ \begin{aligned} |N-q|&\le 2\sqrt{q}\quad \deg f=3\\ \text{and } |N+1-q|&\le 2\sqrt{q}\quad \deg f=4. \end{aligned} \] More generally, let \(f(x,y)\) be a polynomial in \(x\) and \(y\) of total degree \(d>0\) with coefficients in \(\mathbb F_q\), and let \(N\) denote the number of solutions \((x,y)\in \mathbb F_q\times \mathbb F_q\) of the equation \(f(x,y)= 0\). A. Weil (1940) proved that if the polynomial \(f(x,y)\) is absolutely irreducible then we have \[ |N-q|\leq 2g\sqrt{q}+ c_1(d), \tag{1} \] where \(g\) is the genus of the algebraic curve \(f(x,y)=0\) (so that \(g\le (d-1)(d-2)/2)\) and \(c_1(d)\) is a constant depending on \(d\). Weil’s proof depends on algebraic geometry.

Recently, S. A. Stepanov (1969–1974) gave a new and elementary proof of some special cases of Weil’s result which does not depend on algebraic geometry but is related to A. Thue’s method in Diophantine approximations. In particular, Stepanov proved that for \(f(x,y)= y^d-f(x)\) we have \[ |N-q|\leq c_2(d)\sqrt{q} \tag{2} \] with some constant \(c_2(d)\) depending only on \(d\). Later, by the Thue-Stepanov method, E. Bombieri (1973) and W. M. Schmidt (1973) proved (2) in which \(f(x,y)\) is an absolutely irreducible polynomial of total degree \(d\); (1) will follow from (2) by the theory of the zeta function of algebraic curves.

In these Lectures the following generalization of (2) is proved by the method of Stepanov’s. Let \(f(x_1,\ldots, x_n)\) be a polynomial over \(\mathbb F_q\) in \(n\ge 2\) variables \(x_1,\ldots,x_n\) of total degree \(d>0\), and let \(N\) be the number of solutions \((x_1,\dots, x_n)\in \mathbb F_q^n\) of the equation \(f(x_1,\ldots, x_n)= 0\). If \(f(x_1,\ldots, x_n)\) is absolutely irreducible then we have \[ |N-q^{n-1}|\le c_3(d) q^{n-(3/2)}+ c_4(d) q^{n-2} \] with some constants \(c_3(d)\) and \(c_4(d)\) both of which can be explicitly written in terms of \(d\) alone.

The whole presentation is highly clear and readable; several results are proved in more than one way and, as the author says in the Preface, the style adopted is amiably ‘leisurely’.

Let \(f(x)\) be a polynomial in \(x\) with coefficients in a finite field \(\mathbb F_q\) of \(q\) elements. E. Artin (1924) conjectured and H. Hasse (1936) proved that the number \(N\) of solutions \((x,y)\in \mathbb F_q\times \mathbb F_q\) of the equation \(y^2= f(x)\) satisfies \[ \begin{aligned} |N-q|&\le 2\sqrt{q}\quad \deg f=3\\ \text{and } |N+1-q|&\le 2\sqrt{q}\quad \deg f=4. \end{aligned} \] More generally, let \(f(x,y)\) be a polynomial in \(x\) and \(y\) of total degree \(d>0\) with coefficients in \(\mathbb F_q\), and let \(N\) denote the number of solutions \((x,y)\in \mathbb F_q\times \mathbb F_q\) of the equation \(f(x,y)= 0\). A. Weil (1940) proved that if the polynomial \(f(x,y)\) is absolutely irreducible then we have \[ |N-q|\leq 2g\sqrt{q}+ c_1(d), \tag{1} \] where \(g\) is the genus of the algebraic curve \(f(x,y)=0\) (so that \(g\le (d-1)(d-2)/2)\) and \(c_1(d)\) is a constant depending on \(d\). Weil’s proof depends on algebraic geometry.

Recently, S. A. Stepanov (1969–1974) gave a new and elementary proof of some special cases of Weil’s result which does not depend on algebraic geometry but is related to A. Thue’s method in Diophantine approximations. In particular, Stepanov proved that for \(f(x,y)= y^d-f(x)\) we have \[ |N-q|\leq c_2(d)\sqrt{q} \tag{2} \] with some constant \(c_2(d)\) depending only on \(d\). Later, by the Thue-Stepanov method, E. Bombieri (1973) and W. M. Schmidt (1973) proved (2) in which \(f(x,y)\) is an absolutely irreducible polynomial of total degree \(d\); (1) will follow from (2) by the theory of the zeta function of algebraic curves.

In these Lectures the following generalization of (2) is proved by the method of Stepanov’s. Let \(f(x_1,\ldots, x_n)\) be a polynomial over \(\mathbb F_q\) in \(n\ge 2\) variables \(x_1,\ldots,x_n\) of total degree \(d>0\), and let \(N\) be the number of solutions \((x_1,\dots, x_n)\in \mathbb F_q^n\) of the equation \(f(x_1,\ldots, x_n)= 0\). If \(f(x_1,\ldots, x_n)\) is absolutely irreducible then we have \[ |N-q^{n-1}|\le c_3(d) q^{n-(3/2)}+ c_4(d) q^{n-2} \] with some constants \(c_3(d)\) and \(c_4(d)\) both of which can be explicitly written in terms of \(d\) alone.

The whole presentation is highly clear and readable; several results are proved in more than one way and, as the author says in the Preface, the style adopted is amiably ‘leisurely’.

Reviewer: SaburĂ´ Uchiyama (Tsukuba)

### MSC:

11-02 | Research exposition (monographs, survey articles) pertaining to number theory |

11G20 | Curves over finite and local fields |

11G25 | Varieties over finite and local fields |

11T06 | Polynomials over finite fields |

14G10 | Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) |

14G15 | Finite ground fields in algebraic geometry |

11T55 | Arithmetic theory of polynomial rings over finite fields |

11T23 | Exponential sums |

11T24 | Other character sums and Gauss sums |

11D41 | Higher degree equations; Fermat’s equation |