Andrzej Schinzel 80: an outstanding scientific career

Transcription

1 1 / 18 Andrzej Schinzel 80: an outstanding scientific career Kálmán Győry

2 Short biography 2 / 18 Professor Andrzej Schinzel world-famous number theorist, old friend of Hungarian mathematicians. Born on April 5, 1937 at Sandomierz, Poland. MSc in 1958 at Warsaw University. PhD in 1960 from Institute of Mathematics of the Polish Academy of Sciences (PAS),under the guidance of Waclaw Sierpinski. Waclaw Sierpinski

3 Short biography Rockefeller Foundation Fellowship at the University of Cambridge (under the supervision of Harold Davenport) and the University of Uppsala (studied under Trygve Nagell). Habilitation in Associate Professor 1967, Full Professor Corresponding Member, 1994 Ordinary Member of the PAS. A great number of invited lectures all over the world, many Visiting Professorships. Significant activity in international organizations and committees. Andrzej Schinzel in Cambridge, / 18

4 Scientific results, Publications, SELECTA 350 research papers, almost exclusively in number theory, 30 papers while he was an undergraduate student, more than 90 articles since Hungarian coauthors: Erdős (3), Győry (3), Hajdu (2), Pintér (1), Ruzsa (1). 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. About one-third of his papers and two books are devoted to this topic. Selected Topics in Polynomials, University of Michigan, Ann Arbor, Polynomials with Special Regard to Reducibility, Encyclopedia of Math. and Appl. 77, Cambridge University Press, Cambridge, / 18

5 Scientific results, Publications, SELECTA Schinzel s work is very influential in many areas of number theory. On the occasion of his 70th birthday, to honor his scientific accomplishments, SELECTA (2007) containing his 100 most important papers with commentaries by outstanding mathematicians. The themes covered: First volume Diophantine equations and integral forms (R. Tijdeman) Continued fractions (E. Dubois) Algebraic number theory (D. W. Boyd and D. L. Lewis) Polynomials in one variable (M. Filaseta) Polynomials in several variables (U. Zannier) Hilbert s Irreducibility Theorem (U. Zannier) 5 / 18

6 Scientific results, Publications, SELECTA Second volume Arithmetic functions (K. Ford) Divisibility and congruences (H. W. Lenstra, Jr.) Primitive divisors (C. L. Stewart) Prime numbers (J. Kaczorowski) Analytic number theory (J. Kaczorowski) Geometry of numbers (W. M. Schmidt) Other papers (S. Kwapien and E. Szemerédi) Last chapter: unsolved problems and unproved conjectures (proposed by Andrzej Schinzel in the years ). 6 / 18

7 A selection from the most outstanding results of Andrzej Schinzel We present some of the most outstanding results of Andrzej Schinzel, selected mainly from the paper "The work of Andrzej Schinzel in number theory" (1999) by W. Narkiewicz. Arithmetic functions 1954, vast generalization of an earlier result on the Euler s ϕ-function: If α 1,..., α k are given nonnegative reals (including infinity) then there exists a sequence n 1, n 2,... of integers such that lim ϕ(n r + j)/ϕ(n r + j 1) = α j for j = 1,..., k. r Similar assertion and its quantitative version for the sum of divisors function σ(n) with Wang (1958); extension to a large class of positive multiplicative function with Erdős (1960/61). 7 / 18

8 A selection from the most outstanding results of Andrzej Schinzel Prime numbers 1958, together with W. Sierpinski, Schinzel analyzed various unexpected consequences of his celebrated conjecture. Hypothesis H (Schinzel): if f 1,..., f s Z[x] are irreducible polynomials having positive leading coefficients and there is no integer > 1 that is a divisor of f 1 (n) f s (n) for n = 1, 2, 3,..., then for infinitely many positive integers n the values f 1 (n),..., f s (n) are primes. For s = 1, this is a conjecture of Bouniakowski; quantitative formulation: Bateman, Horn (1962); generalized version: Schinzel (1966); relation between H and a conjecture on Diophantine equations with parameters: Schinzel (1980). 8 / 18

9 Polynomials, questions of irreducibility For n 2, formidable task: determine all reducible n-nomials over a field K which is either a number field or a function field in one variable. For binomials Vahlen (1895), K = Q; Capelli (1897), K any field of zero characteristic. For trinomials over a number field K A special case of a celebrated theorem of Schinzel: if n 2m then the trinomial x n + ax m + b (a, b K \{0}) is irreducible it has a linear or quadratic factor or lies in one of fifteen series of polynomials, of which fourteen are described explicitly and the fifteenth is possibly empty. 9 / 18

10 Polynomials, questions of irreducibility Lacunary polynomials Using a result of Bombieri and Zannier, Schinzel (1999) proved the following significant result on lacunary polynomials: Let a = [a 0,..., a k ] (Q ) k+1, n = [n 1,..., n k ] N k with 0 < n 1 <... < n k N, and K = Q(a 1 /a 0,..., a k /a 0 ). Then there is a constant c(a) such that the number of n for which a 0 + a 1 x n a k x n k has more than one noncyclotomic irreducible factor over K is less than c(a)n [(k+1)/2]. The exponent is best possible. This shows that apart from cyclotomic factors, there are "many" irreducible polynomials among the lacunary polynomials under consideration. Schinzel: further outstanding results on the reducibility of trinomials, quadrinomials, lacunary polynomials and polynomials in several variables. 10 / 18

11 Polynomials, questions of irreducibility For P Z[x], denote by P the sum of squares of the coefficients of P. Problem of P. Turán Does there exist a constant C such that for every polynomial f (x) Z[x] there is an irreducible polynomial g(x) Z[x] of degree at most deg f such that f g C? This problem is very difficult. It becomes simpler if one removes the condition that the degree of g should not exceed deg f. 11 / 18

12 Polynomials, questions of irreducibility 12 / 18 The following deep theorem of Schinzel (1970) gives a partial answer to Turán s problem. For any polynomial f (x) Z[x] there exist infinitely many irreducible polynomials g(x) Z[x] such that { 2 if f (0) 0, f g 3 always. One of them, g 0 satisfies deg g 0 < exp{(5 deg f + 7) ( f + 3)}. Later, several related results, among others by Bérczes, Győry and Hajdu.

13 Number of terms of a power of a polynomial 13 / 18 Many people investigated the number of terms of the square of a polynomial. As a substantial improvement of an earlier result of Schinzel (1987), Schinzel and Zannier (2009) proved the following theorem: Let f be an univariate polynomial over a field K of characteristic 0 and s a positive integer. Denote by t and T the number of terms of f and f s. Then provided that T 2. t 2 + log(t 1) log(4s) The same is proved if char K is large with respect to s, deg f, t and T.

14 Genus of primitive binary quadratic forms 14 / 18 In a joint paper with A. Baker (1971), Schinzel proved that every genus of primitive binary quadratic forms of discriminant D represents a positive integer not exceeding c(ε) D (3/8)+ε, ε being an arbitrary positive number. Further, the conjecture is stated that one can replace 3/8 by 0. This conjecture is still open, but by the best known result, due to Heath-Brown, 3/8 can be replaced by 1/4.

15 Diophantine problems Schinzel obtained a great number of significant results on Diophantine equations. Perhaps the most striking result, due to him and Tijdeman, states that if a polynomial P(x) with rational coefficients has at least two distinct zeros and the equation P(x) = y m has a solution in integers m, x and y with y 0, ±1, then m is bounded by an effectively computable constant depending only on P. This implies that a rational polynomial having at least three distinct zeros can represent at most finitely many proper powers. Main tool in the proof: Baker s method. Later, quantitative versions, generalizations and many applications. 15 / 18

16 ACTA ARITHMETICA 16 / 18 As a successor of his teacher W. Sierpinski, Schinzel was the editor in chief of ACTA ARITHMETICA, the first international journal devoted exclusively to number theory, for 39 years, from 1969 to Among the other editors were/are Cassels, Davenport, Erdős, Jarnik, Heath-Brown, Kaczorowski, Linnik, Mordell, Schmidt, Sprindzuk, Tijdeman, Turán and Zannier. They and the other members of the advisory board, including Halász, Pintz, Sárközy and myself from Hungary, have determined the line of the journal. Schinzel and his Acta Arithmetica influenced many mathematicians, stimulating their thinking and mathematical career.

17 Some honors and awards 17 / invited speaker at the International Congress of Mathematicians, Nice International Conference in Zakopane on the occasion of 60th birthday of Andrzej Schinzel, conference volume at the de Gruyter, Honorary Member of the Hungarian Academy of Sciences Publication of Andrzej Schinzel SELECTA at the European Mathematical Society, on the occasion of his 70th birthday Honorary Doctor Degree from the University of Poznan.

18 Happy Birthday Professor Schinzel! 18 / 18