Heritage of European Mathematics. Zürich: European Mathematical Society Publishing House (EMS) (ISBN 978-3-03719-038-8/hbk). xiv, 1393 p. EUR 168.00 (2007).

Nobody is able to present seriously all the works contained in these two volumes except A. Schinzel himself. A. Schinzel wrote his first paper at the age of 17: “{\it Sur la décomposition des nombres naturels en somme de nombres triangulaires distincts}”, Bull. Acad. Polon. Sci., 1954. Most of his first papers were much influenced by his supervisor W. Sierpiński. The central theme of Schinzel’s work is arithmetical and algebraic properties of polynomials in one or several variables, in particular questions of irreducibility and zeros of polynomials; this concerns about one third of his papers and he wrote two books on this subject, each containing a lot of old and new information. The selection presented in these two volumes contains 100 papers chosen among more than 200 papers published by Schinzel, and also a list of unsolved problems and unproved conjectures proposed by Schinzel in the years 1956--2006. This collection is organized into 13 sections, each theme being presented and commented by an expert. I present this list below and for each section I select one result of Schinzel. My main criterion is simplicity and elegance and the second one is novelty and originality, many of these results were in advance when they appeared. But I note that most of the deepest results of Schinzel are very long to state, they contain long lists of explicity special cases and it was impossible to reproduce them here. A. “Diophantine equations and integral forms”, commented by R. Tijdeman -- 10 papers. Reference: “On the equation $y^m=P(x)$” [with {\it R. Tijdeman}], Acta Arith. (1976). -- If a polynomial $P(x)$ with rational coefficients has at least two distinct zeros then the equation $\,y^m=P(x)$, where $x$ and $y$ are integers, $\vert y\vert >1$, implies $\,m<c(P)$ (effective). B. “Continued fractions and integral forms”, commented by E. Dubois -- 3 papers. Reference: “On some problems of the arithmetical theory of continued fractions”, Acta Arith. (1961). -- For a given quadratic surd $\xi$ let $\text{lp}\,\xi$ be the length of the shortest period of the continued fraction expansion of $\xi$, and let $f(x)$ be a polynomial with integer coefficients, degree $d$, and positive leading coefficient $a$, then, if $d$ is odd or if $d$ is even and $a$ is not a square, $$ \limsup \text{lp}\, \sqrt{f(n)} = \infty. $$ C. “Algebraic number theory”, commented by D. W. Boyd and D. J. Lewis -- 10 papers. Reference: “On the product on the conjugates outside of the unit circle of an algebraic number”, Acta Arith. (1973). -- The main result implies: Let $K$ be a CM field ({\it i.e.}, a number field which is either totally real or a totally complex quadratic extension of such a field) of degree $n$ and let $P\in K[X]$ be a polynomial of degree $d$ such that $X^d \overline P(1/X) \not= \text{constant} \cdot P(X)$ then $$ \prod_{i=1}^n \prod_{\vert \alpha_{i,j}\vert >1} \vert \alpha_{i,j}\vert \ge \left({1+\sqrt 5\over 2}\right)^{n/2}, $$ where the $\alpha_{i,j}$ are the roots of the conjugates $P^{(j)}$ of $P$, $j=1$, …, $n$. D. “Polynomials in one variable”, commented by M. Filaseta -- 17 papers. Reference: “Reducibility of lacunary polynomials II”, Acta Arith. (1970). -- For any polynomial $f$ with integer coefficients there exist infinitely many irreducible polynomials $g$ with integer coefficients such that $$ \Vert f-g\Vert \le 3, $$ where $\Vert P\Vert$ denotes the sum of the squares of the coefficients of a polynomial $P$. E. “Polynomials in several variables”, commented by U. Zannier -- 10 papers. Reference: “Reducibility of polynomials of the form $f(x)-g(y)$”, Colloq. Math (1967). -- Let $f$ and $g$ be non-constant polynomials with rational coefficients and let the degree of $f$ be a prime $p$. Then $f(x)-g(y)$ is reducible over the complex field if and only if $g(y)=f(c(y))$ and either $c$ has rational coefficients or $$ f(x)-g(y) = A(x+\alpha)^p-Bd(y)^p, $$ where $d$ has rational coefficients and $A$, $B$ and $\alpha$ are rationals. F. “Hilbert Irreducibility Theorem”, commented by U. Zannier -- 3 papers. Reference: “The least admissible value of the parameter in Hilbert Irreducibility Theorem”, [with {\it U. Zannier}], Acta Arith. (1995). -- Let $F_1$, …, $F_h$ be irreducible polynomials in $\Bbb Q [t,x]$ such that $\deg F_i \le D$ and the height of each $F_i$ is at most $H$, then there exists a rational number $t^*=u/v$ such that each $F_i(t^*,x)$ is irreducible over $\Bbb Q$ and $$ \max\{\vert u\vert , \vert v\vert \}\le \exp\bigl(10^{10}D^{100\,hD^2\log D} (1+\log^2 H) \bigr). $$ G. “Arithmetic functions”, commented by K. Ford -- 6 papers. Reference: “Sur l’équation $\varphi(x)=m$”, Elemente der Math. (1956). -- For any positive integer $n$, there exist infinitely many rational integers $m$ which are multiples of $n$ and such that the equation $\varphi(x)=m$ has no solution. H. “Divisibility and congruences”, commented by H.W. Lenstra jun. -- 11 papers. Reference: “On the congruence $a^x \equiv b\pmod p$”, Bull. Acad. Pol. Sci. (1960). -- If $a$ and $b$ are rational integers, $a>0$ and $b\not= a^k$ ($k$ -- rational integer), then there exist infinitely many prime numbers $p$ for which the congruence $a^x \equiv b\pmod p$ has no solution in rational integers $x$. I. “Primitive divisors”, commented by C. L. Stewart. -- 6 papers. Reference: “On primitive factors of Lehmer numbers II”, Acta Arith. (1963). -- Let $L$ and $M$ be coprime rational integers, suppose that $K=L-4M$ is non-zero, let $\alpha$ and $\beta$ be the roots of the trinomial $z^2-L^{1/2}z+M$ and put $$ P_n = \cases (\alpha^n-\beta^n)/(\alpha -\beta ), & \text{for $n$ odd},\\ (\alpha^n-\beta^n)/(\alpha^2 -\beta^2 ), & \text{for $n$ even}. \endcases $$ Let $e=3$, $4$ or $6$. If $L^{1/2}$ is rational, $K^{1/2}$ is an irrational integer of the field $\Bbb Q(\xi)$, $K$ is divisble by the cube of the discriminant of the field, $\kappa_e=k_e(M)$ is square-free [where for a positive rational integer $x$ the number $k_e(x)$ is equal to $x$ divided by the greatest $e$th power dividing it], $$ \eta_e = \cases 2, & \text{if } e=6,\ M\equiv 3\pmod 4, \\ 1, & \text{otherwise}, \endcases $$ and $n/(\eta_e\kappa_e)$ is an integer relatively prime to $e$, then for $n>n_e(L,M)$ (effectively computable), $P_n$ has at least $e$ primitive factors. J. “Prime numbers”, commented by J. Kaczorowski. -- 5 papers. Reference: “On two theorems of Gelfond and some of their applications”, Acta Arith. (1967). -- If $f(x)$ is any quadratic polynomial without a double root then $$ \liminf_{x \to \infty} {\text{P}(f(x)) \over \log \log x} \ge \cases 4/7, &\text{if $f$ is irreducible}, \\ 2/7, & \text{if $f$ is reducible}, \endcases $$ where $\text{P}(x)$ denotes the greatest prime factor of a non-zero rational integer. K. “Analytic number theory”, commented by J. Kaczorowski. -- 4 papers. Reference: “On an analytic problem considered by Sierpiński and Ramanujan”, in New trends in Probability and Statistics, v. 2, Analytic and Probabilistic Methods in Number Theory (1992). -- Let $r(n)$ be the number of representations of a positive integer $n$ as a sum of two squares, then $$ \sum_{n\le x}r^2(n)=4\,x\log x+cx +\Omega(x^{3/8}). $$ [Note: Sierpiński had proved that $\,\sum_{n\le x}r^2(n)=4\,x\log x+cx +O(x^{3/4}\log x)$ in 1906, and this is the first “$\Omega$” result on this problem.] L. “Geometry of numbers”, commented by W. M. Schmidt. -- 4 papers. Reference: “A decomposition of integer vectors”, [with {\it S. Chał adus}] PLISKA Stud. Mat. Bulgarica (1991). -- For a vector ${\bold n}=(n_1,\dots,n_k)$ put $h({\bold n})=\max \vert n_i\vert $. Then for any non-zero vector ${\bold n}=(n_1,n_2,n_3)$ of rational integers there exist independent vectors ${\bold p}$ and ${\bold q}$ in $\bbfZ^3$ such that ${\bold n}=u{\bold p}+v{\bold q}$, with $u$, $v\in \bbfZ$ and $$ h({\bold p})\cdot h({\bold q}) < \sqrt{ \tfrac43 h ({\bold n}) }. $$ M. “Other papers”, commented by S. Kwapień. -- 5 papers. Reference: “An inequality for determinants with real entries”, Colloq. Math. (1978). -- For every matrix $A=(a_{ij})_{i,j\le n}$ with real entries we have the inequality $$ \vert \det (A)\vert \le \prod_{i=1}^{ n}\max\left\{ \sum_{1\le j\le n, \ a_{ij}>0} a_{ij}, - \sum_{1\le j\le n, \ a_{ij}<0}a_{ij} \right\}. $$ Conjectures. -- We end this list by a very famous conjecture of Schinzel (1958), “conjecture H”: If $f_1$, …, $f_k$ are irreducible univariate polynomials with integer coefficients and positive leading coefficient such that the product $f_1(x) \cdots f_k(x)$ has no fixed divisor ${}>1$, then there exist infinitely positive integers $x$ such that all the numbers $f_i(x)$, $1\le i\le k$, are primes. I hope that this enumeration will show the reader the extraordinary variety of Schinzel’s works and let him guess the incredible amount of information contained in these two volumes.