TY - JOUR
T1 - A proof of van der Waerden’s Conjecture on random Galois groups of polynomials
AU - Bhargava, Manjul
N1 - Funding Information:
This article is the text of the announcement of and talk on this result given on the occasion of Don Zagier’s 70th birthday celebration in the Number Theory Web Seminar on July 1, 2021. Some details and arguments have been added for readability and completeness. Happy birthday, Don!We are extremely grateful to Benedict Gross, Danny Neftin, Andrew O’Desky, Robert Lemke Oliver, Ken Ono, Fernando Rodriguez-Villegas, Arul Shankar, Don Zagier, and the anonymous referee for all their helpful comments and encouragement. Most of all, we thank Don Zagier for his friendship and years of inspiration—wishing him a very happy birthday and many happy returns!.
Publisher Copyright:
© 2023, International Press, Inc.. All rights reserved.
PY - 2023
Y1 - 2023
N2 - Of the (2H +1)n monic integer polynomials f(x) = xn+a1xn−1+···+an with max{|a1|,…,|an|} ≤ H,howmanyhave associated Galois group that is not the full symmetric group Sn? There are clearly ≫ Hn−1 such polynomials, as may be obtained by setting an = 0. In 1936, van der Waerden conjectured that O(Hn−1) should in fact also be the correct upper bound for the count of such polynomials. The conjecture has been known previously for degrees n ≤ 4, due to work of van der Waerden and Chow and Dietmann. In this expository article, we outline a proof of van der Waerden’s Conjecture for all degrees n.∗.
AB - Of the (2H +1)n monic integer polynomials f(x) = xn+a1xn−1+···+an with max{|a1|,…,|an|} ≤ H,howmanyhave associated Galois group that is not the full symmetric group Sn? There are clearly ≫ Hn−1 such polynomials, as may be obtained by setting an = 0. In 1936, van der Waerden conjectured that O(Hn−1) should in fact also be the correct upper bound for the count of such polynomials. The conjecture has been known previously for degrees n ≤ 4, due to work of van der Waerden and Chow and Dietmann. In this expository article, we outline a proof of van der Waerden’s Conjecture for all degrees n.∗.
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U2 - 10.4310/PAMQ.2023.v19.n1.a3
DO - 10.4310/PAMQ.2023.v19.n1.a3
M3 - Article
AN - SCOPUS:85153598584
SN - 1558-8599
VL - 19
SP - 45
EP - 60
JO - Pure and Applied Mathematics Quarterly
JF - Pure and Applied Mathematics Quarterly
IS - 1
ER -