Non-injectivity of Nonzero Discriminant Polynomials and Applications to Packing Polynomials
aa r X i v : . [ m a t h . N T ] F e b Non-injectivity of Nonzero DiscriminantPolynomials and Applications to PackingPolynomials
K˚are Schou GjaldbækFebruary 9, 2021
Abstract
Define the sector S ( α ) := { ( x, y ) ∈ R : 0 ≤ y ≤ αx } . The sectoris called irrational if α is irrational. A packing polynomial on S ( α ) is apolynomial which bijectively maps integer lattice points of S ( α ) onto thenon-negative integers. We show that an integer-valued quadratic poly-nomial on R can not be injective on the integer lattice points of anyaffine convex cone if its discriminant is nonzero. A consequence is thenon-existence of quadratic packing polynomials on irrational sectors of R . In the seminal 1878 paper
Ein Beitrag zur Mannigfaltigkeitslehre [3], Cantorintroduces the polynomial f ( x, y ) = x + ( x + y − x + y + 2)2which bijectively maps N × N onto N , N denoting the positive integers. Trans-lating, and interchanging the variables, leads to the two Cantor Polynomials F ( x, y ) = 12 ( x + y )( x + y + 1) + x ,G ( x, y ) = 12 ( x + y )( x + y + 1) + y . These two quadratics are bijections from N × N onto N , N denoting thenon-negative integers. See Figure 1In 1923, Fueter and P´olya [5] prove that the two Cantor Polynomials arethe only two quadratic polynomials admitting such a bijection. They furtherconjecture that a bijection is impossible for a polynomial of degree other thantwo. Fueter and P´olya’s proof relies on a corollary of the Lindemann-Weierstraßtheorem which says that the trigonometric and hyperbolic functions evaluated125914202735 1481319263443 37121825334252 611172432415162 1016233140506173 1522303949607285 2129384859718498 28374758708397112 013610152128 2471116222937 58121723303847 913182431394858 1419253240495970 2026334150607183 2734425161728497 35435262738598112Figure 1: The two Cantor polynomials correspond to enumerating the latticealong diagonals.at non-trivial algebraic numbers are transcendental. In 2001, Vsemirnov [11]proves the theorem using elementary methods. Theorem 2 of this paper allowsfor Fueter and P´olya’s strategy to work without the need for the theorem ofLindemann-Weierstraß (Fueter and P´olya’s result is a special case of the classi-fication of rational sectors given in [2]).In two papers [7, 8] from 1977, Lew and Rosenberg develop a more gen-eral theory, and some of the terminology they introduce has taken hold. Theyprovide a partial result on Fueter and P´olya’s conjecture in proving the non-existence of polynomials of degree 3 and 4. The general problem remains open.For some preliminary results, Lew and Rosenberg consider polynomials onarbitrary sectors, regions that are the convex hull of two half-lines starting atthe origin, yet they do not study polynomial bijections from general sectorsonto N . This is the subject of Nathanson’s 2014 paper Cantor Polynomialsfor Semigroup Sectors [9]. Nathanson looks at quadratic polynomial bijectionsfrom arbitrary sectors in the first quadrant, in particular the lattice points inthe convex cone spanned by the x -axis and the line y = αx for some α > α = 1 /n , n ∈ N .Furthermore, he finds two quadratic polynomials for each sector with integer α -value. He raises the question of which rational values for α allow for quadraticpolynomial bijections, and whether such is possible for irrational α .The same year, Stanton and Brandt answer the questions regarding rationalsectors. In [10], Stanton determines all quadratic polynomials for α ∈ N . Inaddition to the polynomials discovered by Nathanson, she finds two quadraticpacking polynomials for the sectors given by α = 3 , Let α >
0. Define the sector S ( α ) = { ( x, y ) ∈ R : 0 ≤ y ≤ αx } . We let S ( ∞ ) denote the first quadrant, i.e. the sector pertaining to the origi-nal problem. Adopting the terminology of Lew and Rosenberg, we refer to abijection from S ( α ) ∩ Z onto N as a packing function , or, in the case of apolynomial, which is our sole focus, packing polynomial .An immediate prerequisite for a packing polynomial is that it is integer-valued . That is, it must take integer values on integer lattice points. A conse-quence of standard results on integer-valued polynomials (see e.g. [6], Chp. X,Lem. 6.4) is that a quadratic packing polynomial on any sector must be of theform P ( x, y ) = A x ( x − Bxy + C y ( y − Dx + Ey + F (1)with A, B, C, D, E, F ∈ Z . Lew and Rosenberg make the following observation(see [7], Prop. 3.4). Lemma 1.
Let P ( x, y ) be a packing polynomial on the sector S ( α ) . If ( m, n ) ∈ S ( α ) \ { (0 , } is an integer lattice point, then the homogeneous quadratic partof P ( x, y ) : P ( x, y ) = A x + Bxy + C y , must take only positive values on the ray { ( xm, xn ) , x > } . An immediate consequence is that we must have
A > Let ω , ω ∈ R with ω = ω and define the closed convex cone C ( ω , ω ) = { uω + vω : u, v ≥ } and for ω ∈ R the affine convex cone C ω ( ω , ω ) = C ( ω , ω ) + ω . Theorem 2.
Let P : R → R be an integer-valued quadratic polynomial. If thediscriminant of P is non-zero, then P cannot be injective on the integer latticepoints of any affine convex cone. In fact, it is only required for the polynomial to be an injection into N , a class Lew andRosenberg call storing functions . roof. Let P ( x, y ) have the form (1). We will denote its discriminant ∆ = B − AC and use the shorthand notation D ′ = D − A , E ′ = E − C . Let C = C ω ( ω , ω ) be an arbitrary affine cone. Fix coprime integers r, s with r = 0.Every lattice point lies on a line L ( i ) r,s : y = sr x + ir for a unique i . For each i ,consider the restriction of P to L ( i ) r,s : Q i ( x ) = P (cid:18) x, sr x + ir (cid:19) = 12 r ( Ar + 2 Brs + Cs ) x + 1 r (( Br + Cs ) i + r ( D ′ r + E ′ s )) x + const.If r, s are chosen such that Ar + 2 Brs + Cs = 0, then the values of Q i ( x ) aresymmetric around x i = − ( Br + Cs ) i + r ( D ′ r + E ′ s ) Ar + 2 Brs + Cs . The corresponding y -coordinate on L ( i ) r,s is y i = ( Ar + Bs ) i − s ( D ′ r + E ′ s ) Ar + 2 Brs + Cs . This means that, for any choice of r, s with Ar + 2 Brs + Cs = 0, we have P ( x i + r, y i + s ) = P ( x i − r, y i − s )for all i .These points of symmetry, ( x i , y i ), fall on the straight line, L sym r,s , with slope − Ar + BsBr + Cs which passes through the point( x , y ) = (cid:18) CD ′ − BE ′ ∆ , AE ′ − BD ′ ∆ (cid:19) . (To see this, replace i with (( AE ′ − BD ′ ) r − ( CD ′ − BE ′ ) s ) in the formulasfor x i , y i .) Note that the point ( x , y ) does not depend on the choice of r, s . We want to choose r, s such that ( x i + r, y i + s ) and ( x i − r, y i − s ) are bothlattice points in C for some i . This will violate injectivity.Put C = C ( x ,y ) ( ω , ω ) and pick an arbitrary lattice point ( m, n ) ∈ C ∩ C .Choosing sr = − A ( m − x ) + B ( n − y ) B ( m − x ) + C ( n − y ) When ∆ = 0, P ( x, y ) can be rewritten as P ( x, y ) = A x − x ) + B ( x − x )( y − y ) + C y − y ) + D ′ x + E ′ y + F .
The point ( x , y ) is the center of the level curves P ( x, y ) = n which are ellipses when ∆ < >
4n lowest terms, we have − Ar + BsBr + Cs = n − y m − x . This means that L sym r,s passes throughthe lattice point ( m, n ) ∈ C ∩ C and therefore infinitely many since its slopeis rational. So for infinitely many i , ( x i , y i ) is a lattice point in C ∩ C andeventually we will find an i for which both ( x i + r, y i + s ) and ( x i − r, y i − s ) arelattice points in C . L ( r , s ) L ( r , s ; i ) ( x , y ) ω ( m, n ) = ( x i , y i )( m − r, n − s ) ( m + r, n + s ) CC Figure 2: L ( r, s ) with rs = − A ( m − x )+ B ( n − y ) B ( m − x )+ C ( n − y ) passing through a lattice pointin C ∩ C . Let P ( x, y ) be a quadratic packing polynomial on the sector S ( α ) with discrim-inant ∆. S ( α ) is a (affine) convex cone, so by the previous section we musthave ∆ = 0. Employing the strategy of Lew and Rosenberg [7], we consider theregions R n = { ( x, y ) ∈ S ( α ) : 0 ≤ P ( x, y ) ≤ n } . For a packing polynomial, each region R n contains n + 1 lattice points. Thismeans that it is a necessary condition thatlim n →∞ (cid:18) n R n ∩ Z ) (cid:19) = 1 . (2)Furthermore, we have 5 emma 3. If P ( x, y ) is a quadratic packing polynomial on S ( α ) , then R n ∩ Z ) = area( R n ) + O ( √ n ) , as n → ∞ .Proof. Since the discriminant is zero, the level curves P ( x, y ) = n are parabolasand either R n is bounded for all n or all level curves, including for negative n , fall inside S ( α ) which is impossible if P is a packing polynomial. We cantherefore apply a theorem of Davenport ([4], p. 180) to estimate the number oflattice points in each region R n . We have | area( R n ) − R n ∩ Z ) | ≤ h ( | π x ( R n ) | + | π y ( R n ) | ) + h , where | π x ( R n ) | and | π y ( R n ) | denotes the lengths of the projections onto the x -and y -axis, respectively, and h is a fixed constant. The length of y -projectionis bounded by the level curves intersection with the line y = αx or the topmostpoint on the parabola. The length of x -projection is bounded by the levelcurves intersection with the x -axis, the line y = αx or the rightmost point onthe parabola. Either is O ( √ n ) as n → ∞ .At this point, we want to note that, since B = AC , the homogeneousquadratic part of P ( x, y ), P ( x, y ) = 12 ( √ Ax ± √ Cy ) , is non-negative and vanishes only on the rational line Ax + By = 0. By Lem. 1,this means that P ( x, y ) is strictly positive inside S ( α ) except at the origin. Lemma 4. If P ( x, y ) is a quadratic packing polynomial on S ( α ) , then area( R n ) = n Z arctan α dθ A cos θ + B cos θ sin θ + C sin θ + O ( √ n ) as n → ∞ .Proof. Switching to polar coordinates, the equations of the level curves take theform r p ( θ ) + rp ( θ ) + F = n , where p ( θ ) = A cos θ + B cos θ sin θ + C sin θ and p ( θ ) = D ′ cos θ + E ′ sin θ .So r = O ( √ n ) and r = np ( θ ) + O ( √ n ). If A denotes the (possibly empty) areaof S ( α ) bounded by the level curve P ( x, y ) = 0, then the area of R n is given byarea( R n ) = 12 Z arctan α r dθ − A = 12 Z arctan α np ( θ ) dθ + O ( √ n ) . The constant h denotes the maximum number of disjoint intervals one can obtain fromintersecting R n with a line parallel to one of the coordinate axes. Our only concern is that itis bounded by some value. heorem 5. If P ( x, y ) is a quadratic packing polynomial on S ( α ) , then α = A − B .
Proof.
Using Lem. 3 and 4, we can calculate the limit from (2) by computingthe integrallim n →∞ (cid:18) n R n ∩ Z ) (cid:19) = 12 Z arctan α dθ A cos θ + B cos θ sin θ + C sin θ = Z α dtA + 2 Bt + Ct , applying the variable change t = tan θ . Since, by Thm. 2, B = AC , we eitherhave B = C = 0, in which case Z α dtA + 2 Bt + Ct = Z α dtA = αA , or Z α dtA + 2 Bt + Ct = 1 C Z α dθ (cid:0) t + BC (cid:1) = 1 B − αC + B .
As noted above, this must be equal to 1 if P is a packing polynomial. Solvingfor α , we get the desired result.An immediate consequence of Thm. 5 is the following. Corollary 6.
There are no irrational sectors allowing for quadratic packingpolynomials.
References [1] M. Brandt. Quadratic packing polynomials on sectors of R . arXiv:1409.0063v1 , 2014.[2] M. Brandt and K. Gjaldbæk. Classification of quadratic packing polyno-mials on sectors of R . In preparation , 2021.[3] G. Cantor. Ein Beitrag zur Mannigfaltigkeitslehre.
Journal fur die reineund angewandte Mathematik , 84:242–258, 1878.[4] H. Davenport. On a principle of Lipschitz.
Journal of the London Mathe-matical Society , 1(3):179–183, 1951.[5] R. Fueter and G. P´olya. Rationale Abz¨ahlung der Gitterpunkte.
Vierteljschr. Naturforsch. Ges. Z¨urich , 58:380–386, 1923.[6] S. Lang.
Algebra . Springer, 2002. 77] J. S. Lew and A. L. Rosenberg. Polynomial indexing of integer lattice-points I. General concepts and quadratic polynomials.
Journal of NumberTheory , 10(2):192–214, 1978.[8] J. S. Lew and A. L. Rosenberg. Polynomial indexing of integer lattice-points II. Nonexistence results for higher-degree polynomials.
Journal ofNumber Theory , 10(2):215–243, 1978.[9] M. B. Nathanson. Cantor polynomials for semigroup sectors.
Journal ofAlgebra and its Applications , 13(5), 2014.[10] C. Stanton. Packing polynomials on sectors of R . Integers , 14, 2014.[11] M. A. Vsemirnov. Two elementary proofs of the Fueter-P´olya theorem onpairing polynomials.