There should be more math! This could be mathier! -- Buffy Summers, channeling Rupert Giles, BtVS Episode 2.8 ("The Dark Age")
| An explicit
approach to Hypothesis H for polynomials over finite
fields Schinzel's Hypothesis H predicts that a family of irreducible polynomials over the integers satisfying certain necessary local conditions simultaneously assumes prime values infinitely often. Here we consider an analogue of Hypothesis H for one-variable polynomials over the q-element finite field Fq and show that it holds whenever q is large compared to the degree of the product of the polynomials involved. We also show that for fixed q, the conclusion of our Hypothesis H holds for "almost all" single-polynomial families. Along the way we propose a new polynomial analogue of the Hardy-Littlewood/Bateman-Horn conjectures. |
To appear in: The anatomy of integers. Proceedings of a conference on the anatomy of integers, Montreal, March 13th-17th, 2006 eds: J.M. de Koninck, A. Granville and F. Luca | |
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On a conjecture of Beard, O'Connell and West concerning
perfect polynomials (joint w/ L. Gallardo and O.
Rahavandrainy) Following Beard, et al., we call a polynomial over a finite field Fq perfect if it coincides with the sum of its monic divisors. The study of perfect polynomials was initiated by Carlitz's doctoral student Canaday in the case q=2, who proposed the still unresolved conjecture that every perfect polynomial over F2 has a root in F2. Beard, O'Connell and West later proposed the analogous hypothesis for all finite fields. Counterexamples to this general conjecture were found by Link (in the cases q=11, 17) and Gallardo & Rahavandrainy (in the case q=4). Here we show that the Beard-O'Connell-West conjecture fails in all cases except possibly when q is prime. When q=p is prime, utilizing a construction of Link we exhibit a counterexample whenever p ≡ 11 or 17 mod 24. On the basis of a polynomial analog of Schinzel's Hypothesis H, we argue that if there is a single perfect polynomial over the finite field Fq with no linear factor, then there are infinitely many. Lastly, we prove without any hypothesis that there are infinitely many perfect polynomials over F11 with no linear factor. | To appear in Finite Fields and their Applications | |
| Remarks on
Hypothesis H and an impossibility theorem of Ram Murty Dirichlet's 1837 theorem that every coprime arithmetic progression a mod m contains infinitely many primes is often alluded to in elementary number theory courses but usually proved only in special cases (e.g., when m=3 or m=4), where the proofs parallel Euclid's argument for the existence of infinitely many primes. It is natural to wonder whether Dirichlet's theorem in its entirety can be proved by such `Euclidean' arguments. In 1912, Schur showed that one can construct an argument of this type for every progression a mod m satisfying a2 ≡ 1 mod m, and in 1988 Murty showed that these are the only progressions for which such an argument can be given. Murty's proof uses some deep results from algebraic number theory (in particular the Chebotarev density theorem). Here we give a heuristic explanation for this result by showing how it follows from Bunyakovsky's conjecture on prime values of polynomials. We also propose a widening of Murty's definition of a Euclidean proof. With this definition it appears difficult to classify the progressions for which such a proof exists. However, assuming Schinzel's Hypothesis H, we show that again such a proof exists only when a2 ≡ 1 mod m. | In preparation | |
| Simultaneous
prime specializations of polynomials over finite fields Let n be a positive integer and let f1(T), ..., fr(T) be pairwise nonassociated irreducible polynomials over a finite field Fq with the degree of f1*...*fr bounded by B. We show that the number of univariate monic polynomials h of degree n for which all of f1(h(T)), ..., fr(h(T)) are irreducible over Fq is qn/nr + On,B(qn-1/2) provided gcd(q,2n)=1. As an application, fix an infinite arithmetic progression a mod m and fix pairwise nonassociated irreducibles f1(T), ..., fr(T) over Fp with the degree of f1*...*fr bounded by B. If p is sufficiently large depending only on m, r, and B, then there are infinitely many monic polynomials h(T) with deg h ≡ a mod m and all of f1(h(T)), ..., fr(h(T)) irreducible over Fp. | Submitted | |
| A
polynomial analogue of the twin prime conjecture We consider the problem of counting the number of (not necessarily monic) `twin prime pairs' P, P+M in Fq[T] of degree n, where M is a polynomial of degree ≤ n. We formulate an asymptotic prediction for the number of such pairs as qn tends to infinity and then prove an explicit estimate confirming the conjecture in those cases where q grows faster than n2. When M has degree n-1, our theorem implies the general validity of a result claimed by Hayes in 1963, but proven only under additional hypotheses. When M has degree zero, our theorem refines a result of Effinger, Hicks & Mullen. |
To appear in the Proceedings of the AMS | |
| Arithmetic
properties of polynomial specializations over finite
fields We present applications of some recent results that establish a partial finite field analogue of Schinzel's Hypothesis H. For example, we prove that the distribution of gaps between degree n prime polynomials over Fp is close to Poisson for p large compared to n. We also estimate the number of polynomial substitutions without prime factors of large degree ("smooth" polynomial substitutions); this confirms a finite field analogue of a conjecture of Martin in certain ranges of the parameters. Other topics considered include an analogue of Brun's constant for polynomials and "smooth" values of neighboring polynomials. |
Submitted |
| Solved and Unsolved Problems in Elementary Number
Theory (Ross Program 2005) An eclectic collection of of wonderful, wild and wacky problems, conjectures and theorems from the wide, wide world of elementary arithmetic. | |
| Rational (!)
Cubic and Biquadratic Reciprocity (Ross Program 2005, Princeton Grad Student Seminar, Dartmouth Number Theory Seminar) Overview of a cubic reciprocity law for rational primes published by Jacobi in 1827. The version we present here is from Z.-H. Sun's paper "On the theory of cubic residues and nonresidues." | |
| Some Special
Cases of an Fq[u]-variant of Hypothesis H (2005 Anatomy of Integers Workshop, Montréal) Overview of the results in the paper "Irreducibility Preserving Substitutions for Polynomials over Finite Fields." (See below.) | |
| Is There a Pattern in the Primes? (2006 Dartmouth Exploting Math Program) Various patterns in the primes are examined in order to elucidate the ways in which primes can be said to behave randomly. (Ulam's prime spiral and related graphics were produced using code by Michel Charpentier. The remaining illustrations were shamelessly stolen from the Prime Pages, Dartmouth's Chance newsletter and Richard Guy's article on Conway's prime-producing machine.) |
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| Integers and Polynomials: Parallel Universes (Dartmouth Grad Student Seminar -- August 2006) One of the main themes of modern number theory is the strong analogy between number fields (finite extensions of Q) and global function fields (finite extensions of the field of rational functions over a finite field). We offer a down-to-earth view of this subject by looking at some easy-to-state problems in number theory and their analogues in the ring of polynomials over a finite field. Particular attention is paid to the study of perfect numbers (vs. "perfect polynomials") and the distribution of primes. | |
| Simultaneous
Prime Values of Polynomials in Positive Characteristic (2007 AMS/MAA Joint Meetings -- invited talk for the Special Session on the Arithmetic of Function fields) Overview of the results in the paper "Irreducibility Preserving Substitutions for Polynomials over Finite Fields" and "Counting Irreducibility-Preserving Substitutions for Polynomials over Finite Fields." (See below.) | |
| Prime
Polynomial Patterns (April 26 2007; Quebec-Vermont Number Theory seminar -- the latter half was also presented at thhe April 14-15 AMS Eastern Sectional Meeting in Hoboken, NJ) The first half of this talk is intended as a gentle introduction to the analogies between integers and polynomials for graduate students with basic knowledge of number theory. The latter half surveys the results written up in the aforementioned papers as well as recent, related work not yet written up. | |
| Arithmetic properties of polynomial
specializations over finite fields (September 30; Maine-Quebec number theory conference) Summarizes my work on polynomials, with a special emphasis on the paper of the same name. Since it was a twenty-minute talk, the subset of slides used in the actual talk is a small (but cofinite) subset of the slides in the PDF file. |
| E. Landau's proof of the Big Picard theorem If f has an essential singularity at z, then f assumes every value with at most one exception in every deleted neighborhood of z. The exposition follows the book Complex Analyis: An Invitation by Rao and Stetkaer, which I highly recommend generally. | Short Proofs of the three Sylow Theorems Proofs of the three Sylow theorems via double-coset counting. I am not quite sure of the origins of these arguments; they are certainly not with me. Nevertheless, I am happy to advertise them, as they deserve to be better known. F. Lemmermeyer points out that F. Lorenz's book Algebra (Volume 1: Fields and Galois Theory) takes a similar approach to the Sylow theorems. |
| An Elementary Proof of Dirichlet's Theorem in
the Polynomial Setting We translate H.N. Shapiro's proof of Dirichlet's theorem on primes in progressions into the setting of the ring of polynomials over a finite field. |
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| A Proof of
the Sylvester-Schur Theorem A product of k consecutive integers each larger than k is always divisible by a prime greater than k (e.g., taking the integers in (k,2k] we recover Bertrand's postulate). We largely follow the treatment of Narkiewicz's Classical Problems in Number Theory but correct a numerical error. | |
| An Analogue of
Goldbach's Conjecture for certain Polynomial Rings Various authors (e.g., Hayes, Rattan & Stewart, Betts) have noted that the analog of Goldbach's conjecture is true for the ring of polynomials with integer coefficients: every nonconstant polynomial is a sum of two irreducibles. In t his note we prove the same for more general rings of polynomials, including, e.g., the ring of polynomials in any number of variables over a Noetherian domain with infinitely many maximal ideals. In particular we get the result for the ring of polynomials in at least two variables over any finite field. The situation in one variable is still not understood, but see this paper of Hayes and its errata. (Also, the paper `A polynomial analogue of the twin prime conjecture' listed below can be considered a sequel of sorts to the Hayes paper.) |
| Not Always
Buried Deep: Selections from Analytic and Combinatorial Number
Theory A textbook on elementary/analytic/combinatorial number theory, a huge part of which was written for my senior thesis under Matt Baker at the University of Georgia. Intended for publication someday, but only after extensive revision. As it stands it is recommended not only to budding number theorists but to students of logic. (I'm not a specialist in the latter area, but I have a hunch that the number of errors in this book bears some relation to the theory of inaccessible cardinals.) | On hiatus | PDF DVI jDVI |