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24 FEATURES Rubel’s Problem: from Hayman’s List to the Chabauty Method
EDWARD CRANE AND GENE S. KOPP
Walter Hayman’s list Research Problems in Function Theory has been updated for its 50th anniversary. Problem 4.27, posed by Lee Rubel, was related to the famous Prouhet-Tarry-Escott problem. Some modern tools of computational number theory can be used to nd explicit counterexamples.
Walter Hayman’s list a problem in function theory but rather a problem in number theory. It was posed by the American Research Problems in Function Theory [9] is a analyst Lee Rubel (1927–1995), whose work spanned collection of problems collated by Walter Hayman in di erential equations, approximation theory, and the 1967 and fondly known as Hayman’s list. It gave a big theory of analog computing. Among many other impetus to the development of geometric function surprising results, Rubel showed that there is a single theory, the part of complex analysis concerned with entire function whose derivatives are dense in the the geometric properties of analytic functions. space of entire functions with respect to the topology of locally uniform convergence. Here is problem 4.27, Walter loved to as it appears in [9]: encourage younger mathematicians. He and his wife Margaret 4.27 (Lee Rubel). Let f (x) be a real polynomial founded the British of degree n in the real variable x such that Mathematical Olympiad, f (x) = 0 has n distinct (real) rational roots. and the mathematics Does there necessarily exist a (real) non-zero genealogy website lists number t such that f (x) − t = 0 has n distinct his 20 PhD students (real) rational roots? and 133 mathematical (I can prove this for n = 1,2,3.) descendants. His own research was You might like to try the case n = 3 for yourself. groundbreaking, particul- Walter Hayman. arly his work on the Photo credit: MFO The phrasing suggests Rubel hoped for an a rmative Bieberbach conjecture answer, extrapolating from the low-degree cases. and in the area of Nevanlinna theory, which relates the growth and covering properties of meromorphic In a straw poll of our local number theorists at the functions. He was awarded the LMS De Morgan medal University of Bristol, all had the opposite intuition to in 1995. Fifty years after the publication of the list, Rubel’s. We’ll give a calculation-free proof below that ≥ Walter Hayman teamed up with Eleanor Lingham the answer is no, at least for even degrees n 8. (the editor of this newsletter) to gather the state What is more interesting is to try to exhibit explicit ≥ of the art on each of the problems. The resulting counterexamples. For each degree n 4 we pose Fiftieth Anniversary Edition [10] was published in 2019. the following challenge: We were sorry to learn that Walter Hayman passed away on January 1st 2020. Problem Rubel(n): Exhibit f ∈ ℚ[x] of degree n such that f − t has n distinct rational roots if and only if t = 0. Rubel’s problem If f is a polynomial of degree n, t ≠ 0, and both The fourth chapter of Hayman’s list is about f and f − t have n distinct rational roots, then we polynomials. Among the problems there, number can rescale the roots of f and f − t to obtain an 4.27 stands out for the strange reason that it is not ideal solution of the Prouhet-Tarry-Escott problem (see
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inset). Those are hard to nd, but this doesn’t mean rst case f − t cannot have two distinct real roots, it is easy to prove that any particular polynomial f and in the second it cannot have two distinct 2-adic is a solution of Rubel(n). roots. Ruling out repeated roots stops Rubel’s problem from Prouhet–Tarry–Escott problem being trivial. For example f = x (x − 1)2 (x + 1)2 has a local minimum at 1 and a local maximum at −1, so Given k,n ∈ ℕ, nd two distinct sets of f − t has at most three real roots if t ≠ 0. integers A and B, both of size n, such that The constraint that f has n distinct rational roots Õ i Õ i makes the problem a lot more fun. It implies that a = b for i = 1,...,k. for K = ℝ or K = ℚ , every su ciently small a ∈A b ∈B p perturbation f˜ of f still factors completely over K . A simple appplication of the pigeonhole This is because each root of f can be perturbed to principle shows that solutions exist whenever a nearby root of f˜ in K . This is shown in the p-adic n > k (k +1)/2. A solution is called ideal when case by Hensel’s lemma. So we cannot hope to solve n = k + 1. In that case we have Rubel(n) by nding a local obstruction. Ö Ö (x − a) − (x − b) = c a ∈A b ∈B
for some non-zero constant c. The largest k A non-constructive solution of Rubel’s problem for which an ideal solution is known is 11. This solution was found in 1999 by Nuutti Kuosa, We will answer Rubel’s problem by showing that for Jean-Charles Meyrignac and Chen Shuwen: each n ≥ 4 , there exists a solution to Rubel(2n). A = {±35,±47,±94,±121,±146,±148}, Choose rationals 0 < a1 < ··· < an such that the polynomial f = În (x −a2) has n−1 distinct critical B = {±22,±61,±86,±127,±140,±151} . i=1 i values. This condition holds generically in An (ℚ). For this example, we have Case 1: f is a solution of Rubel(n). 2 c = 67440294559676054016000 Then F := f (x ) is a solution of Rubel(2n), since it has the 2n distinct roots ±a ,...,±a . However, if = 212.39.53.72.112.132.17.19.23.29.31 . 1 n F −t has 2n distinct rational roots for some nonzero − If you can prove that there is no ideal t, then f t must have n distinct rational roots, solution of Prouhet-Tarry-Escott with k ≥ 12, contrary to the hypothesis. then you will also have proved that every Case 2: f is not a solution of Rubel(n). ≥ polynomial of degree n 13 with n distinct Dene G (x,w) = (f (x) − f (w))/(x − w). Thinking Rubel( ) rational roots is a solution of n . See of G as a polynomial in x, its coe cients are Borwein [5] for more information about the polynomials in w, so its discriminant h belongs to Prouhet-Tarry-Escott problem. ℚ[w]. The roots of h in ℂ are the n − 2 non-critical preimages under f of each of the n − 1 critical values of f . Each of these is a simple root of h. So h has degree d = (n − 1)(n − 2) and has no No local obstructions repeated complex root. In particular, d ≥ 6, and the discriminant of h is non-zero. It is easy to write down an f ∈ ℚ[x] of degree n ≥ 3 for which there is no t ∈ ℚ such that f − t If f − t has rational roots q1 < ··· < qn, we have 2n has n distinct roots. For instance, this is true for rational points on the hyperelliptic curve Y 2 = h(X ), f = x3 + x, because it maps ℝ to ℝ injectively. given by For another example, f = x3 − 2x is an injective mapping from ℚ to ℚ. If x ≠ y, but f (x) = f (y), then x2 + xy + y 2 = 2, but for x,y ∈ ℚ, not both © Ö ª q , ± (q − q )® , k = 1,...,n . zero, the valuation of x2 + xy + y 2 at 2 is always even. k i j ® i < j ® These are both examples of local obstructions. In the « i,j ≠k ¬
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Since disc(h) ≠ 0, this is a smooth a ne curve over ℚ of genus g = d(d − 2)/2e = (n − 1)(n − 2)/2 − 1. Faltings’ theorem Although its closure in the projective plane is not smooth, it can be embedded in a smooth curve In 1922 Louis Mordell proved the seminal result in projective 3-space, by a map that takes rational that the abelian group of rational points on points to rational points. Faltings’ theorem says that any elliptic curve dened over ℚ is nitely any smooth projective curve of genus at least 2 has generated. At the end of same paper, he only nitely many rational points (see inset). It follows conjectured a restricted version of what that the curve Y 2 = h(X ) has only nitely many became known as the Mordell conjecture, that rational points. any smooth algebraic curve dened over ℚ with genus at least 2 has only nitely many We deduce that there are only nitely many rational rational points. − values of t for which f t has n distinct rational The Mordell conjecture was proved by Gerd < ··· < roots. Let t1 tm be the complete sorted list Faltings in 1983, and he won a Fields Medal of these rational values. Since f is not a solution of in 1986 for this work. A few years later, Rubel( ) ≥ n , m 2. Consider the degree 2n polynomial Paul Vojta gave a very di erent proof using Diophantine approximation and Arakelov F := (f − t1)(f − t2) . intersection theory, and Enrico Bombieri soon By construction F has 2n distinct rational roots. We gave a more elementary version of this proof. claim F is a solution of Rubel(2n). Suppose F −t has Faltings’ theorem is ine ective: it does not 2n distinct rational roots for some t ∈ ℚ\{0}. Let U1 tell us how to enumerate all of the rational and U2 be the roots of the quadratic (x−t1)(x−t2)−t, points. Bombieri’s proof in principle gives so that U1 + U2 = t1 + t2 and a computable bound on how many rational points a curve has but no bound on how much F − t = (f − U1)(f − U2) . ink it takes to write the points down. At most n of the roots of F − t can be roots of f − Ui , for i = 1,2, so exactly n are roots of each. In particular U1 and U2 are rational and they An explicit example for Rubel(7) belong to {t3,...,tm }. But because of the sorting + + this contradicts U1 U2 = t1 t2. Consider the polynomial This proof was simple enough, but we used the big f = (x − 4)(x − 3)(x − 1)x (x + 1)(x + 3)(x + 4) . hammer of Faltings’ theorem to crack the nut! And we haven’t yet written down any explicit counterexample Because f is odd, the discriminant of f − t with to Rubel’s problem. respect to x is an even polynomial in t. We write disc(f − t) = h(t 2), where h is the irreducible cubic h = − 823543 t 3 + 353645809920 t 2 A sextic f such that f − t has six rational roots − 35639879984676864 t for innitely many t + 95144698561167360000.
Dene four rational functions of u: The sextic h(t 2) has six real roots, which are the ∈ ℚ − 2u2 + 6u + 1 3u2 + 2u − 2 critical values of f . Any t for which f t has A = , B = , seven roots q ,...,q ∈ ℚ gives us a rational point u2 + u + 1 u2 + u + 1 1 7 2 u − 4u − 3 2 Ö C = , and R = (ABC ) . (X ,Y ) = ©t, (qi − q j )ª 2 + + ® u u 1 1≤i< j ≤7 Then « ¬ on the a ne curve C : Y 2 = h(X 2). x6 − 14x4 + 49x2 − R = (x2 − A2)(x2 − B 2)(x2 −C 2) . The curve C is called bi-elliptic because it has Our argument using Faltings’ theorem does not apply nonconstant maps to two di erent elliptic curves. 2 to this case because for each choice of u the critical First, C covers the elliptic curve E1 : Y = h(X ) , by values of this sextic polynomial in x are repeated. the map (X ,Y ) 7→ (X 2,Y ). This map does not help
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us, because the point of C coming from t = 0 has innite order in E1, so the group of rational points E1 (ℚ) is innite. Second, C covers the elliptic curve
2 3 E2 : Y = X h(1/X ) .
Any rational point (t,y) on C such that t ≠ 0 yields a nite rational point
(L-r) Gerd Faltings, Robert F. Coleman. Photo credit: MFO 2 3 (X ,Y ) = (1/t ,y/t ) ∈ E2 (ℚ) .
A calculation in Magma shows that E2 (ℚ) is trivial, The Chabauty method consisting only of the point at innity. Hence C has no nite point (t,y) with t ≠ 0, and f − t never To give an explicit solution to Rubel(6), we will use factors completely over ℚ for t ∈ ℚ \{0}. Chabauty’s method to enumerate the rational points on a smooth a ne curve of the form y 2 = disc(f −t), where deg(f ) = 6. When f has no repeated critical Professor Elmer Rees CBE value, this curve has an embedding dened over ℚ (1941–2019) into a smooth projective curve in ℙ3.
Ed Crane writes: Rubel’s problem was the Chabauty’s method originated as a proof of the last mathematical question that I discussed Mordell conjecture for a restricted class of curves [7], with my friend Elmer Rees, whom I got to over 40 years before the work of Faltings. Chabauty’s know when he was the rst director of the method was made e ective in the 1980s through Heilbronn Institute for Mathematical Research. Robert F. Coleman’s work on p-adic integration. Although he was an algebraic topologist, The Chabauty–Coleman method is a technique for Elmer had worked before on problems about bounding the number of rational points on a smooth the factorization of polynomials over the curve C by embedding them in the p-adic points integers. In Oxford in the 1970s, he wrote on the Jacobian variety J = Jac( C). The Jacobian A criterion for a brief note with Walter Feit, J = Pic0 ( C) is an abelian variety of dimension g , a polynomial to factor completely over the where g is the genus of C. The Abel-Jacobi map is integers. They proved this criterion in order a rational embedding uP : C → J dened with to establish a result about the algebraic 0 respect to a base point P0. A point P on C is sent topology of complex algebraic varieties. Later, to the divisor class uP (P ) = [P − P0]. in Edinburgh, Elmer and Chris Smyth proved 0 some necessary divisibility properties of If p is a prime, base change gives an embedding solutions of the Prouhet-Tarry-Escott problem J(ℚ) → J(ℚp ). The p-adic closure of J(ℚ) in 0 [14]. Even though Elmer was already unwell J(ℚp ) is a p-adic submanifold. Its dimension r = when we talked about Rubel’s problem, it dim J(ℚp ) is always bounded above by the rank sparked his characteristic enthusiasm. r = rkJ(ℚ) of the rational points of the Jacobian as an abelian group. Another solution of Rubel(7) is By composing with the Abel-Jacobi map, we have an 0 embedding C(ℚ) → J(Q p ). When r < g , Chabauty f = (x − 3)(x − 2)(x − 1)x (x + 1)(x + 2)(x + 3) . used the properties of this embedding to show there are only nitely many points on the curve. To prove this one can use the method of Victor In fact, an explicit bound may be given, as was Flynn and Joe Wetherell [8] to enumerate the rational shown by Robert F. Coleman. The proof and full points on the bi-elliptic curve y 2 = disc(f − t). Their statement of Coleman’s theorem requires a type method is elementary, using ideas from the proof of of p-adic integration now called Coleman integration, the Mordell-Weil theorem, but we omit the details. that treats degree 0 divisors as paths of integration
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and allows one to extend the denition of p-adic This is a hyperelliptic curve of genus 2, so Chabauty integration outside the radius of convergence of the can be applied if the rank of the Jacobian is 0 or 1. standard denition. We state Coleman’s theorem for = ( ) = ( ) easy reference (Theorem 5.3 in [13]); see [13] and the The rank r rkJ ℚ of the Jacobian J Jac C is references therein for the denition of the Coleman found using a 2-descent algorithm, which computes integral and other notation and terms. the rank of the 2-part of the Selmer group to give an upper bound, combined with a search for points Theorem (Coleman). Let Cbe a curve of genus g ≥ 2 to give a lower bound. Both are implemented by over ℚ, and let J = Jac( C). Let p be a prime of good the Magma function RankBounds. To reduce the reduction for C, let r = rk J(ℚ), and let r 0 ≤ r be runtime of RankBounds, we rst compute a minimal the dimension of the closure of J(ℚ) in J(ℚp ) as a Weierstrass model for C, given by the equation p-adic manifold. Assume that r 0 < g . y 2 − (x + 1)y 0 1 = 17915904x5 − 51347635x4 − 621566x3 (a) Let l be a nonzero 1-form in H ( Cℚp ,Ω ) with 0 2 the property that, if Qi ,Qi ∈ C(ℚp ) such that + − + 0 108253979x 92802025x 22173442. Í 0 Í ∫ Qi (Q − Qi ) ∈ J (ℚ), then l = 0. Such i i i Qi an l necessarily exists, and we may assume by When applied to this model, RankBounds returns a scaling that it reduces to a nonzero 1-form l ∈ lower bound of 0 and an upper bound of 1. H 0 ( C ,Ω1). Suppose Q ∈ C(F ), and let m = Fp p The Jacobian has a nontrivial rational point u∞ (P ) = − ordQ l. If m < p 2, then the number of points [P − ∞] coming from the known point P = ( ) + in C ℚ reducing to Q is at most m 1. (0,68040) ∈ C(ℚ) coming from the factorisation of f . The torsion group of the Jacobian is (b) If p > 2g , then # C(ℚ) ≤ # C(Fp ) + (2g − 2). computed to be trivial with the Magma function TorsionSubgroup ( ) 0 ; thus, u∞ P has innite order. In the case when r < g (in particular, when r < g ), So, rkJ(ℚ) = 1. part (b) of Coleman’s theorem gives an upper bound on the number of rational points on C. Part (a) may The Magma function Chabauty is then used to be used to give a narrower upper bound in many enumerate the rational points on C. This function cases. However, this bound is not always tight. implements the Mordell-Weil sieve as described by Bruin and Stoll [4] to rule out conjugacy classes In practice, it seems that the Chabauty-Coleman modulo various primes until the Coleman bounds are method can be used — in combination with made tight (under the assumption that rkJ(ℚ) = 1). a technique called the Mordell-Weil sieve — to For our curve C, Magma does a Mordell-Weil sieve enumerate with proof all the rational points on a using local information at the primes {19,37,41} and curve when r < g . When the algorithm terminates, gives the full set of rational points on C as it yields a tight upper bound, but it has not been proven that it always terminates. C(ℚ) = {(0,−68040), (0,68040),∞}.
Thus, f −t does not factor completely over ℚ, except A solution of Rubel(6) when t = 0. In fact, this method shows that, for t ≠ 0, the Galois group of the polynomial f − t is never We now show that the following polynomial is a contained in the alternating group A6. solution to Rubel(6): A challenge: Can you solve Rubel(4) or Rubel(5)? 5 f = (x − 2)(x − 1)x (x + 1)(x + 2) x + . 2 Current developments of Chabauty’s method Let C be the curve dened by the equation The past few years have seen the development of 2 2 y = 8 disc(f − t) new variations on Chabauty’s method that allows = 2985984t 5 + 38231885t 4 − 161118396t 3 one to go beyond the r < g regime (or even r 0 < g ). The far-reaching theory of non-abelian − 2 + + 811349595t 1302526656t 4629441600. Chabauty developed by Minhyong Kim allows one
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to replace the Jacobian variety with any one of a [7] C. Chabauty, Sur les points rationnels des collection of non-abelian analogues [11]. Balakrishnan courbes algébriques de genre supérieur à l’unité, and Dogra have rened a special case of Kim’s ideas C. R. Acad. Sci. Paris, 212 (1941), 882–885. into an e ective method called quadratic Chabauty. [8] E. V. Flynn and J. Wetherell, Finding Rational They have used quadratic Chabauty to enumerate Points on Bielliptic Genus 2 Curves. Manuscripta the rational points on specic curves, including math. 100 (1999), 519–533. hyperelliptic curves with (g ,r ) = (2,2) [1] and (g ,r ) = [9] W. K. Hayman, Research Problems in (2,3) [2], and, jointly with Müller, Tuitman, and Vonk, Function Theory, Athlone Press, London (1967). to a non-hyperelliptic curve with (g ,r ) = (3,3) [3]. [10] W. K. Hayman, and E. F. Lingham, Research Problems in Function Theory – Fiftieth Meanwhile, classical Chabauty continues to pay o Anniversary Edition, Springer (2019). in a wide array of problems. An application to a [11] M. Kim, The Unipotent Albanese Map and problem in uid dynamics may be found in [12]; Selmer Varieties for Curves. Publ. Res. Inst. Math. Lemma 18 therein provides a fully worked example Sci., 45 (2009) no. 1, 89–133. of the Chabauty-Coleman method. [12] G. Kopp, The arithmetic geometry of resonant Rossby wave triads. SIAM J. Appl. Alg. Geom., 1 (2017) no. 1, 352–373. Acknowledgements [13] W. McCallum and B. Poonen, The method of Chabauty and Coleman, in Explicit methods The computations in this article were carried out in in number theory, vol. 36 of Panor. Synthèses, the Magma Computational Algebra System [6] using Soc. Math. France, Paris (2012), 99–117. algorithms developed by many people. In particular, [14] E. Rees and C. Smyth, On the constant in the Chabauty program was primarily developed by the Tarry-Escott problem, in Cinquante ans de Michael Stoll with contributions from others, the polynômes/ Fifty years of polynomials, Lecture RankBounds program was written by Brendan Creutz, Notes in Math. Science 1415, Springer (1990) and the elliptic curve rank algorithms that we used are based on the mwrank program of John Cremona. Edward Crane Ed is a senior Heilbronn FURTHER READING Research fellow at the University of Bristol. He [1] J. S. Balakrishnan and N. Dogra, Quadratic started out in geometric Chabauty and rational points, I: p-adic heights. function theory and Duke Math. J., 167 (2018) no. 11, 1981–2038. now works mainly in [2] J. S. Balakrishnan and N. Dogra, Quadratic probability theory. Ed is Chabauty and rational points II: Generalised trying to eliminate bindweed from his vegetable plot, height functions on Selmer varieties. Int. Math. so he spends time trying to locate roots even when Res. Not., rnz362 (2017), 1–86. he is not thinking about polynomials. [3] J. S. Balakrishnan, N. Dogra, J. S. Müller, J. Tuitman, and J. Vonk, Explicit Chabauty–Kim Gene S. Kopp for the split Cartan modular curve of level 13. Ann. of Math., 189 (2019) no. 3, 885–944. Gene is a Heilbronn [4] N. Bruin and M. Stoll, The Mordell–Weil Research fellow at the sieve: proving non-existence of rational points University of Bristol. on curves. LMS J. Comput. Math., 13 (2010), His main interest is 272–306. number theory. He [5] P. Borwein, The Prouhet–Tarry–Escott is especially attracted Problem. In: Computational Excursions in to problems with an Analysis and Number Theory. CMS Books in elementary statement and to interplays between Mathematics, Springer, New York, NY (2002). algebra and analysis. Gene also enjoys recreational [6] W. Bosma, J. Cannon, and C. Playoust, The probability theory (playing Dungeons & Dragons) and Magma algebra system. I. The user language. recreational knot theory (baking soft pretzels). Computational algebra and number theory (London, 1993). J. Symbolic Comput., 24 Photos of the authors by Chrystal Cherniwchan. (1997), 235–265.
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