Classification of Quadratic Packing Polynomials on Sectors of \mathbb{R}^2
aa r X i v : . [ m a t h . N T ] F e b Classification of Quadratic Packing Polynomialson Sectors of R Abstract
We study quadratic polynomials giving bijections from the integer lat-tice points of sectors of R onto N , called packing polynomials. Wedetermine all quadratic packing polynomials on rational sectors. Thisgeneralizes results of Stanton [9], Nathanson [7], and Fueter and P´olya[3]. Introduction
In the 1870s, Cantor introduces the notion of comparing the sizes of sets usingbijective correspondences [2]. He observes that the polynomial f p x, y q “ x ` p x ` y ´ qp x ` y ´ q remarkable property that it represents each positive integer exactlyonce when x and y run over all positive integers. In [3], Fueter and P´olya provethat two derived Cantor polynomials F p x, y q “ p x ` y qp x ` y ` q ` x, G p x, y q “ p x ` y qp x ` y ` q ` y (1)are the only quadratic polynomials which bijectively map N ˆ N to N . Theauthors also formulate the Fueter-P´olya Conjecture which states that the Cantorpolynomials are the only polynomials of any degree which admit such a bijection.Lew and Rosenberg developed the theory further and studied the Fueter-P´olya Conjecture in the 1970s [5, 6]. They prove that polynomial bijectionsof degrees 3 and 4 are impossible, but the general conjecture remains open.They introduce the term packing function for a function which bijectively mapsa dicrete set to the non-negative integers. For some preliminary results, they bemerkenswerthe Eigenshaft , in the words of the author. Adjusted to include 0, one is obtained from the other by swapping coordinates. Lew andRosenberg [5] use the term
Cantor polynomials . Other authors (e.g. Smory´nski [8]) use theterm
Cantor pairing functions . Fueter and P´olya leave them unnamed. sectors of R , which are sets tp x, y q P R | y ď αx, x ě , y ě u . Allowing for α “ 8 , we may think of the first quadrant asa special case of this.Nathanson studies quadratic packing polynomials on the lattice points ofsectors of R in [7]. He finds two packing polynomials for each sector with α “ m , m P N . He shows that F m p x, y q “ p x ´ p m ´ q y qp x ´ p m ´ q y ´ q ` x ` p ´ p m ´ qq y,G m p x, y q “ p x ´ p m ´ q y qp x ´ p m ´ q y ` q ` x ` p´ ´ p m ´ qq y are the only two quadratic packing polynomials on sectors of this type. For eachsector defined by α “ n , n P N , he finds two packing polynomials F n p x, y q “ n x p x ´ q ` x ` y, and G n p x, y q “ n x p x ` q ` x ´ y, (2)but does not rule out the possibility of more. Nathanson’s article ends with alist of open problems: Question 1.
Are there packing polynomials on sectors with irrational α ? Question 2.
Are there packing polynomials of higher degree on sectors?
Question 3.
Classify quadratic packing polynomials on sectors with α P Q .Question 1 is settled in the quadratic case by Gjaldbæk [4], while Question 2remains open. Stanton [9] studies Question 3. She provides a necessary formfor the homogeneous quadratic part of the polynomial and uses it to classifyall quadratic packing polynomials for integral α . Shortly thereafter, Brandt [1]finds a method to tackle the general rational case. The article is incomplete andremained in preprint, but the method is fruitful and the conclusions correct, aswe shall see. In this article, we provide a complete classification of quadraticpacking polynomials on sectors with rational slope (Theorem 11), thus answeringQuestion 3 and completing the classification of quadratic packing polynomialson sectors. As special cases we recover the integral case (Stanton [9]), thereciprocals (Nathanson [7]) and the original scenario (Fueter-P´olya [3]). In this section we define sectors and quadratic packing polynomials. Then, westate Stanton’s necessary form for a quadratic packing polynomial, and providean independent proof of this result.For two linearly independent vectors ω , ω P R , define the sector S p ω , ω q as the conical hull of ω and ω . That is S p ω , ω q : “ t t ω ` t ω : t , t ě u . S p ω , ω q is the convex hull of the rays tω , tω , t ě
0. We willrestrict our attention to sectors where one of the rays is the positive x -axis. Aslong as one of ω or ω is rational, this does not present any loss of generality,as we explain in Section 5. We denote for α ą S p α q Ă R as S p α q : “ S ` p , q , p , α q ˘ “ tp x, y q P R : 0 ď y ď αx u and the integer lattice points in S p α q as I p α q : “ S p α q X Z .S p8q and I p8q will denote the first quadrant and its lattice points, respectively.We will refer to sectors as integral, rational, irrational according to α beingintegral, rational, irrational. Definition 1.
Let I Ă R be an enumerable set. A function f : R Ñ R is called a packing function on I if it maps I bijectively onto N . If f is apolynomial it is called a packing polynomial . Remark.
In Lew and Rosenberg’s definition the set I is always a subset of Z m and Nathanson uses this definition as well. Our broader definition is practicalfor the treatment we present here. If we speak of a packing polynomial on S p α q the enumerable set implied will be I p α q .In this paper we will classify all quadratic packing polynomials (QPPs) onthe lattice points of sectors S p α q . We will determine which values α allow forQPPs and provide formulas for each. Example 2.
Let α “ {
3. Then the polynomial 2 x ´ xy ` y ` y is a quadraticpacking polynomial on S p { q , see Figure 1.Our starting point is the following result (see [4], Equation 1 and Theorem 5). Theorem 3 ([4]) . If P p x, y q is a QPP on I p α q , then P p x, y q “ A x p x ´ q ` Bxy ` C y p y ´ q ` Dx ` Ey ` F with A, B, C, D, E, F P Z . Furthermore, we must have A ą , B “ AC and α “ A ´ B .
Stanton gives the coefficients of the homogeneous quadratic part of P p x, y q in terms of the slope of the sector and a necessary condition on the slope [9]. Theorem 4 ([9]) . Let α “ nm be rational with gcd p m, n q “ . If P p x, y q is aQPP on I ` nm ˘ , then n | p m ´ q and P p x, y q “ n ˆ x ´ m ´ n y ˙ ` ´ D ´ n ¯ x ` ˆ E ´ p m ´ q { n ˙ y ` F with D, E, F P Z . S p { q given in Example 2.This was proved using the Lindemann-Weierstraß Theorem. Using Theo-rem 3, we are able to present a short alternative proof. Proof.
By Theorem 3, α “ nm “ A ´ B , so n p ´ B q “ mA. Since m and n are coprime, we have n | A . Since AC “ B by assumption, nAC “ nB “ n p ´ p A { n q m q “ n ´ Am ` p A { n q Am , so A | n . Since A ą
0, we conclude that A “ n . It follows that B “ ´p A { n q m “ ´ m and C “ B { A “ p ´ m q { n . That C is an integer provides the condition n | p m ´ q . Remark.
The case of I p8q corresponds to n “ m “ k -stairs and Sector Transformation In this section, we introduce two tools that we will use to prove our main result.The first is the notion of staircases , which help us describe the structure ofquadratic packing polynomials. Next, we describe how to transform sectors sothat we can more readily leverage the structure provided by staircases.4he lattice I ` nm ˘ can be subdivided into disjoint sets of lattice points thatfall on the same line with slope nm ´ (or vertical lines in the case m “ l “ gcd p m ´ , n q . We call J i : “ ! p x, y q P I ´ nm ¯ : p m ´ q y “ nx ´ li ) the i th staircase , noting that I ´ nm ¯ “ ğ i “ J i . We refer to the lattice points as steps on the staircase with the first step being theone with the minimal y -coordinate. For consecutive steps p x , y q and p x , y q on a staircase, we have p x , y q “ p x ` p m ´ q{ l, y ` n { l q . The difference of P p x, y q evaluated on consecutive steps, regardless of the staircase, is P p x ` p m ´ q{ l, y ` n { l q ´ P p x, y q“ ´ D ´ n ¯ p m ´ q{ l ` ˆ E ´ p m ´ q { n ˙ n { l : “ k. We will now transform sectors so that the staircases have a nicer structure.For an invertible 2 ˆ M we study the transformed sector M p S p α qq instead of S p α q . If f p x, y q is a packing function on I p α q , then f p M ´ p x, y qq isa packing function on M p I p α qq . Likewise, any packing function on the trans-formed lattice corresponds to a packing function on the original. Consider thetransformation M “ ˆ ´ m ´ n ˙ , M ´ “ ˆ m ´ n ˙ . This transformation skews the sector and the lattice points with it, turning allstaircases vertical, see Figure 2. Note that in the case of integral sectors thestaircases are already vertical and the transformations are the identity. Thesector S p8q is sent to S p q , so there is really no need to treat S p8q as a specialcase. We will use the following notation:ˆ S ´ nm ¯ : “ M ´ S ´ nm ¯¯ “ S p n q , ˆ J i : “ M p J i q “ "ˆ in { l , y ˙ P ˆ S ´ nm ¯ : pp m ´ q{ l q y ” ´ i p mod n { l q * , ˆ I ´ nm ¯ : “ M ´ I ´ nm ¯¯ “ ğ i “ ˆ J i . Nathanson [7] used transformations of the form ˆ m ˙ to obtain equivalences betweenthe sector S p8q and sectors S ` m ˘ . J i is given by J i “ p l { n q i ` J n { l | i K , where J n { l | i K “ " n { l does not divide i, n { l divides i. The vertical staircases simplify computations. We haveˆ P p x, y q “ P ` M ´ p x, y q ˘ “ n x p x ´ q ` Dx ` kn { l y ` F. (3)ˆ P p x, y q is a QPP on ˆ I ` nm ˘ if and only if P p x, y q is a QPP on I ` nm ˘ . J J J J ¨ ¨ ¨ M ˆ J ˆ J ˆ J ˆ J ¨ ¨ ¨ I ` nm ˘ ˆ I ` nm ˘ Figure 2: Transforming the sector S ` nm ˘ . Example showing nm “ . In this section, we prove a series of lemmas that describe the structure of QPPson transformed sectors. These lead to Proposition 8, which gives the necessaryform of ˆ P p x, y q .Recall that k denotes the difference P p x , y q´ P p x , y q of P on consecutivestairs. Initially, we will assume that k ą
0. Later, in the proof of Theorem 11,we will see that the case k ă y i denote the y -coordinateof the first step on J i . Note that in the case of integral sectors much of the6 i ` c p i q n { l , ¯ y c p i q ¯ ˆ J p n { l q i ˆ J p n { l q i ` c p i q p i, ni q Figure 3: Next staircase with the same congruence class mod k after a staircasewith integral x -coordinate. ˆ P p i, ni q ” ˆ P ´ i ` c p i q n { l , ¯ y c p i q ¯ p mod k q .discussion below could be greatly simplified, since the sector transformation istrivial and ¯ y i “ i . Lemma 5.
Let ˆ P p x, y q be a QPP on ˆ I ` nm ˘ with k ą . Then ˆ P ˆ kn { l , ¯ y k ˙ “ F ` k, and D “ n { l ´ p kl ´ n q ´ ¯ y k . Proof.
First, note that ¯ y p n { l q i “ y p n { l q i ` j “ ¯ y j for all i, j . For each i , let c p i q be the smallest positive integer such that (see Figure 3)ˆ P ´ ˆ J p n { l q i ¯ ” ˆ P ´ ˆ J p n { l q i ` c p i q ¯ p mod k q . The notation is justified by the fact that all values on the same staircasebelong to the same congruence class modulo k . We claim that we can choosearbitrarily large i such that c p i q “ k (in fact, we will see that c p i q “ k for all i large enough). Assume that this is not the case. Then we can choose arbitrarilylarge i with c p i q ą k . For ˆ P to be a packing polynomial, it is necessary thatfrom a certain i , ˆ P evaluated at the last step of ˆ J p n { l q i is exactly k less than7he value at the first step of ˆ J p n { l q i ` c p i q . If that were not the case, then therewould be values missing from the congruence class in question. In other words,ˆ P p i, ni q “ ˆ P ˆ i ` c p i q n { l , ¯ y c p i q ˙ ´ k for all i ě i . Furthermore, we haveˆ P p i, ni q “ ˆ P p i, q ` kli and ˆ P ˆ i ` c p i q n { l , ¯ y c p i q ˙ “ ˆ P p i, q ` ˆ P ˆ c p i q n { l , ¯ y c p i q ˙ ´ F ` c p i q li. In combination, we conclude that for i ą i ˆ P ˆ c p i q n { l , ¯ y c p i q ˙ “ k ` F ` il p k ´ c p i qq . This is impossible if i can be chosen arbitrarily large with c p i q ą k , since oth-erwise P p x, y q would take negative values which cannot happen for a packingpolynomial. We conclude that for large enough i , we must have c p i q “ k . Inturn, we find that ˆ P ˆ kn { l , ¯ y k ˙ “ k ` F, and expanding ˆ P ´ kn { l , ¯ y k ¯ leads to D “ n { l ´ p kl ´ n q ´ ¯ y k . Lemma 6.
Let ˆ P p x, y q be a QPP on ˆ I ` nm ˘ with k ą . Then for all i , ˆ P ´ ˆ J i ¯ ” ˆ P ´ ˆ J i ` k ¯ p mod k q . Proof.
For any i , pick ´ in { l , ¯ y i ¯ and ´ i ` kn { l , ¯ y i ` k ¯ as representatives of pointsfrom ˆ J i and ˆ J i ` k , respectively. Applying the formulas from the previous lemma,a direct calculation showsˆ P ˆ i ` kn { l , ¯ y i ` k ˙ “ ˆ P ˆ in { l , ¯ y i ˙ ` k ` k ˆ il { n ` ¯ y i ` k ´ p ¯ y i ` ¯ y k q n { l ˙ . In the integral sector case, all stairs begin at 0. Else, we have¯ y i ` k ” ´p i ` k q lm ´ ” ¯ y i ` ¯ y k p mod n { l q , the lemma follows. 8 in { l , ¯ y i ¯ ˆ J i ´ k ˆ J i ´ i ´ kn { l , l p i ´ k q ¯ ˆ ˆ P p x , y q ˆ P p x , y q ` k Figure 4: The value of ˆ P p x, y q is k more on the first step of ˆ J i than on the laststep of ˆ J i ´ k . The lines x “ i ´ kn { l and y “ nx intersect at ´ i ´ kn { l , l p i ´ k q ¯ . Lemma 7.
Let ˆ P p x, y q be a QPP on ˆ I ` nm ˘ with k ą . Then ¯ y k “ n { l ´ . Note that this is trivially true for integral sectors.
Proof.
By Lemma 6, for large enough i the value of ˆ P on the first step of ˆ J i is k more than the value on the last step of ˆ J i ´ k . The line x “ i ´ kn { l intersects theline y “ nx at the y -coordinate l p i ´ k q (see Figure 4), soˆ P ˆ i ´ kn { l , l p i ´ k q ˙ ` k ě ˆ P ˆ in { l , ¯ y i ˙ . By direct computation, using the formulas of Lemma 5, we findˆ P ˆ i ´ kn { l , l p i ´ k q ˙ ` k ´ ˆ P ˆ in { l , ¯ y i ˙ “ kn { l p ¯ y k ´ ¯ y i q . According to the above inequality this is non-negative. We conclude that ¯ y k “ n { l ´
1, since this is the highest value a first step can have.
Proposition 8. If ˆ P p x, y q is a QPP on ˆ I ` nm ˘ with k ą and l “ gcd p n, m ´ q ,then we must have n { l | l and ˆ P p x, y q “ n x ˆ x ´ kn { l ˙ ` x ` kn { l y ` F, (4)9 here k ” p m ´ q{ l p mod n { l q .Proof. The necessary form of ˆ P p x, y q follows directly from the initial form inEquation (3) and Lemmas 5 and 7. The condition k ” p m ´ q{ l p mod n { l q follows from Lemma 7 and the fact that ¯ y i p m ´ q{ l ” ´ i p mod n { l q . Again,this is trivial for integral sectors. In this section we prove our main result, Theorem 11. First, we give two morelemmas. Lemma 9 gives a necessary condition for ˆ P to be a packing polynomial.Lemma 10 gives the constant term F . Lemma 9.
Let m, n P N , l “ gcd p m ´ , n q and ˆ P p x, y q “ n x ˆ x ´ kn { l ˙ ` x ` kn { l y ` F, where k ” p m ´ q{ l p mod n { l q . ˆ P p x, y q is a QPP with k ą on ˆ I ` nm ˘ if andonly if " ˆ P ˆ in { l , ¯ y i ˙ : i “ , , . . . , k ´ * “ t , , . . . , k ´ u . Proof.
By Lemma 5, for all i we have that ˆ J i ` k is the next staircase after ˆ J i withvalues in the same congruence class modulo k . Therefore, if ˆ P p x, y q takes thevalues 0 , , . . . , k ´ k ´ P p x, y q is a QPP on ˆ I ` nm ˘ ifˆ P ˆ i ` kn { l , ¯ y i ` k ˙ ´ ˆ P ˆ in { l , ¯ y i ˙ “ k ´ J i ¯ . Applying formula (4), a direct computation showsˆ P ˆ i ` kn { l , ¯ y i ` k ˙ ´ ˆ P ˆ in { l , ¯ y i ˙ “ k ˆ p l { n q i ` ` ¯ y i ` k ´ ¯ y i n { l ˙ . Since ¯ y i ` k ´ ¯ y i p mod n { l q ” ´ p mod n { l q we have ¯ y i ` k ´ ¯ y i “ p n { l q J n { l | i K ´ Lemma 10. If ˆ P p x, y q is a QPP with k ą on ˆ I ` nm ˘ , then F “ p l { n qp k ´ qp k ` q . (5) Proof.
By Lemma 9, we have k ´ ÿ i “ ˆ P ˆ in { l , ¯ y i ˙ “ k ´ ÿ i “ i, l { n k ´ ÿ i “ i ` ˆ ln ´ k l { n ´ ˙ k ´ ÿ i “ i ` kn { l k ´ ÿ i “ ¯ y i ` kF “ . Observing that ¯ y i ` ¯ y k ´ i ” ´ p mod n { l q which implies that ¯ y i ` ¯ y k ´ i “ n { l ´ y “
0, by rearranging the terms the formula k ´ ÿ i “ ¯ y i “ $’’’’’&’’’’’% p k ´ q{ ÿ i “ p ¯ y i ` ¯ y k ´ q , k odd k { ´ ÿ i “ p ¯ y i ` ¯ y k ´ q ` ¯ y k { , k even ,/////./////- “ p k ´ qp n { l ´ q . The formula for F now follows by applying the well-known k ´ ÿ i “ i “ p k ´ q k p k ´ q k ´ ÿ i “ i “ p k ´ q k . Theorem 11.
Let m, n P N with gcd p m, n q “ and put l “ gcd p m ´ , n q . If n | l and p q p m ´ q{ l ” ˘ p mod n { l q , set k “ ˘ , or p q p m ´ q{ l ” ˘ p mod n { l q and l { n “ , set k “ ˘ , or p q p m ´ q{ l ” ˘ p mod n { l q and l { n “ , set k “ ˘ , then P p x, y q “ n ˆ x ´ m ´ n y ˙ ˆ x ´ m ´ n y ´ kln ˙ ` x ` kl ´ p m ´ q n y ` | k | ´ is a QPP on I ` nm ˘ . These are the only QPPs on sectors S p α q Ď R .Proof. Assume that n { l | l and let ˆ P p x, y q have the necessary form in Equation(4). Assume first that k ą
0. We go over the cases k “ , , , and k ą p m ´ q{ l ” k p mod n { l q (again trivial in the integral case, as n { l “ P takes the values 0 , , . . . , k (in any order) on the first steps of the first k staircases. If that is the case, then Lemma 9 guarantees a packing polynomial. Case: k “ F “
0. Sinceˆ P p , q “ , the requirements of Lemma 9 are met.11 ase: k “ F “ l { n ą
0, we must have l { n “ F “
1, by Lemma 10. Since p m ´ q{ l ” p mod n { l q , n { l is odd and we havein the non-integral sector case ¯ y ” ´ p m ´ q{ l ” ´ p mod n { l q , so ¯ y “ n { l ´ .We find that ˆ P p , q “ P ˆ n { l , n { l ´ ˙ “ . By Lemma 9, ˆ P p x, y q is a QPP on ˆ I ` nm ˘ . Case: k “ F “ , F “ l { n “ F “ l { n “
3. Since p m ´ q{ l ” p mod n { l q , we have n { l ı p mod 3 q and 3¯ y ” ´ p mod n { l q and 3¯ y ” ´ p mod n { l q . This means that¯ y “ n { l ´ if n { l ” p mod 3 q n { l ´ if n { l ” p mod 3 q , and ¯ y “ p n { l ´ q if n { l ” p mod 3 q n { l ´ if n { l ” p mod 3 q . In either case, we haveˆ P p , q “ " ˆ P ˆ n { l , ¯ y ˙ , ˆ P ˆ n { l , ¯ y ˙* “ t , u . Again, Lemma 9 verifies that ˆ P p x, y q is a QPP on ˆ I ` nm ˘ . Case: k ą l { n p k ` q ď
12, since 0 ď F ď k ´ ď k ď
11. For k “ l { n “ k ą l { n “
1. Since F is an integer, checking each possible valuefor k and l { n in the formula (5) leaves as the only options k “ , k must be odd. Since k ” p m ´ q{ l p mod n { l q , this means that both2, l { n and n { l have inverses modulo k and thus likewise for l “ p l { n qp n { l q and n . Therefore ˆ P ˆ in { l , ¯ y i ˙ “ n in { l ˆ in { l ´ kn { l ˙ ` in { l ` kn { l ¯ y i ` F ” l { n i ` in { l ` F p mod k q” l { n ˆ i ` l ˙ ´ n ` F p mod k q . p p p p p p p I ` ˘ ˆ I ` ˘ ˇ I ` ˘ “ ˆ I ` ˘ I ` ˘ ˆ ´ ˙ ˆ ´ ˙ ˆ ˙ Figure 5: A QPP with k negative corresponds to a QPP with k positive. Thesectors are not necessarily the same.For any j ” ´ l p mod k q , we haveˆ P ˆ j ` n { l , ¯ y j ` ˙ ” ˆ P ˆ j ´ n { l , ¯ y j ´ ˙ p mod k q . This is impossible by Lemma 6.We have now determined all possibilities for QPPs with k ą
0. We will dealwith the negative cases by showing that there is a one-to-one correspondencebetween QPPs with k “ k and QPPs with k “ ´ k .Assume that ˆ P p x, y q is a QPP on ˆ I ` nm ˘ with k ă
0. By Theorem 4 andEquation (3), we must have n { l | l and ˆ P p x, y q must have the formˆ P p x, y q “ n x p x ´ q ` Dx ` kn { l y ` F with D, F P Z . The involutory transformation L : p x, y q ÞÑ p x, nx ´ y q maps S p n q onto S p n q while vertically flipping the coordinates. See Figure 5. Specifically,the effect on the lattice isˇ I ´ nm ¯ : “ L ´ ˆ I ´ nm ¯¯ “ " p x, y q P S p n q : x “ in { l , pp m ´ q{ l q y ” i p mod n { l q , i P N * . Then ˇ P p x, y q “ ˆ P p x, nx ´ y q “ n x p x ´ q ` p D ` kl q x ´ kn { l y ` F
13s a QPP on ˇ I ` nm ˘ and the difference in values of consecutive steps is ´ k . Fromthe above analysis, we have¯ y ´ k “ n { l ´ ” ´ k p m ´ q{ l p mod n { l q , so p m ´ q{ l ” k p mod n { l q and ˇ P p x, y q “ n x ˆ x ´ ´ kn { l ˙ ` x ` ´ kn { l ` F, where either ´ k “ F “ , or ´ k “ , l { n “ , and F “ , or ´ k “ , l { n “ , and F “ . This means thatˆ P p x, y q “ ˇ P p x, nx ´ y q “ n x ˆ x ´ kn { l ˙ ` x ` kn { l y ` p´ k q ´ . So, regardless of the sign of k , if k “ ˘ , ˘ , ˘ ” p m ´ q{ l p mod n { l q , thenˆ P p x, y q “ n x ˆ x ´ kn { l ˙ ` x ` kn { l y ` | k | ´ I ` nm ˘ and these are the only options. Transforming back to theoriginal integral lattice sector, we find that P p x, y q “ n ˆ x ´ m ´ n y ˙ ˆ x ´ m ´ n y ´ kln ˙ ` x ` kl ´ p m ´ q n y ` | k | ´ I ` nm ˘ . Example 12.
Since 3 ∤ p ´ q , there are no QPPs on the sector I ` ˘ . Example 13.
Let α “ . Then we have n “ m “
5, so n | p m ´ q . Then l “ gcd p m ´ , n q “ n { l “ p m ´ q{ l “ ” ˘ , ˘ p mod 2 q , but l { n “ ‰
3. So only k “ ˘ P p x, y q “ ˆ x ´ y ˙ ˆ x ´ y ´ ˙ ` x,P p x, y q “ ˆ x ´ y ˙ ˆ x ´ y ` ˙ ` x ´ y are the only QPPs on I ` ˘ , see Figure 6.14 31 14842 33231595 6046342416106 95776147352517117 1381169678624836261812 0 72 221361 453221125 7659443120114 115947558433019103 1621371149374574229189Figure 6: The quadratic packing polynomials on S p { q given in Example 13. Example 14.
Let α “ . Then we have n “ m “
7, so n | p m ´ q .Then l “ gcd p m ´ , n q “ n { l “ p m ´ q{ l “ ” ˘ , ˘ p mod 2 q , and l { n “
3. So setting k “ ˘ , ˘ P p x, y q “ ˆ x ´ y ˙ ˆ x ´ y ´ ˙ ` x,P p x, y q “ ˆ x ´ y ˙ ˆ x ´ y ` ˙ ` x ´ yP p x, y q “ ˆ x ´ y ˙ ˆ x ´ y ´ ˙ ` x ` y ` P p x, y q “ ˆ x ´ y ˙ ˆ x ´ y ` ˙ ` x ´ y ` I ` ˘ , see Figure 7. In fact, S p q , S p q and S ` ˘ are theonly sectors (up to equivalence of transformation) with four quadratic packingpolynomials. Example 15.
Consider the integral sectors. That is, n P N , m “
1. Then n | l “ gcd p , n q “ n , n { l “ p m ´ q{ l “ ” ˘ , ˘ , ˘ p mod 1 q . So setting k “ ˘ n . As l { n “ n , for n “ n “ n , the polynomials (2) discovered by Nathanson [7] are QPPs.Additionally, for n “ P p x, y q “ x p x ˘ q ` x ¯ y `
215 41 201152 4833211263 8867493422137 0 103 321992 6647311881 11287654630177 2 01 10534 322113867 6649352416119 2 187 4629154 86634326121 138109836040239 k “ k “ ´ k “ k “ ´ S p { q given in Example 14.and for n “ P p x, y q “ x p x ˘ q ` x ¯ y ` k “ k “ ´ S p q given in Example 15. Example 16.
For S p8q , we set n “ m “ | p´ q .We have l “ gcd p´ , q “ n { l “ p m ´ q{ l “ ´ ” ˘ , ˘ , ˘ p mod 1 q . So setting k “ ˘ l { n ‰ ,
4. We recover the originalCantor polynomials in Equation (1), see Figure 9.16259142027354454 14813192634435364 371218253342526375 6111724324151627487 101623314050617386100 152230394960728599114 2129384859718498113129 0136101521283645 24711162229374656 581217233038475768 9131824313948586981 14192532404959708295 202633415060718396110 2734425161728497111126 k “ k “ ´ S p8q given in Example 16. We now describe two open areas for future research.The Fueter and P´olya problem stated in Question 2, about the the existenceof packing polynomials of higher degree on sectors, remains largely open. On S p8q it is still open for degrees higher than 4, and it is indeed also an openproblem for general sectors.We now pose a question about general irrational sectors. Question 4.
Are there packing polynomials on sectors S p ω , ω q where ω and ω are both irrational?In this article we have focused on sectors of the type S p α q rather than thegeneral type given by two vectors S p ω , ω q . If one of ω or ω is rational,then the situations are equivalent, which we can see as follows. Assume that ω “ p r, s q is rational, by which we mean r, s P N , gcd p r, s q “
1. We canthen find integers a, b P Z such that ar ` bs “
1. The matrix ˆ a b ´ s r ˙ hasdeterminant 1 and sends p r, s q to p , q . If the transformation sends ω outsidethe first quadrant, we can apply transformations ˆ ´ ˙ and ˆ m ˙ , m P Z ,to accommodate that. So, if either of ω or ω is rational, the quadratic packingpolynomials on S p ω , ω q are known. If both ω and ω are irrational, then itis unknown whether packing polynomials are possible.17 eferences [1] M. Brandt. Quadratic packing polynomials on sectors of R . arXiv:1409.0063v1 , 2014.[2] G. Cantor. Ein Beitrag zur Mannigfaltigkeitslehre. Journal fur die reineund angewandte Mathematik , 84:242–258, 1878.[3] R. Fueter and G. P´olya. Rationale Abz¨ahlung der Gitterpunkte.
Vierteljschr.Naturforsch. Ges. Z¨urich , 58:380–386, 1923.[4] K. Gjaldbæk. Non-injectivity of nonzero discriminant polynomials and ap-plications to packing polynomials. arXiv:2102.03392 , 2020.[5] J. S. Lew and A. L. Rosenberg. Polynomial indexing of integer lattice-pointsI. General concepts and quadratic polynomials.
Journal of Number Theory ,10(2):192–214, 1978.[6] J. S. Lew and A. L. Rosenberg. Polynomial indexing of integer lattice-pointsII. Nonexistence results for higher-degree polynomials.
Journal of NumberTheory , 10(2):215–243, 1978.[7] M. B. Nathanson. Cantor polynomials for semigroup sectors.
Journal ofAlgebra and its Applications , 13(5), 2014.[8] C. Smory´nski.