The Proof of a Conjecture For a Continuos Golumb Ruler Model

Preprint

Article

The Proof of a Conjecture For a Continuos Golumb Ruler Model

Altmetrics

Downloads

233

Views

Comments

Tianyu Liu^*,

Cuicui Luo

Tianyu Liu^*,

Cuicui Luo

This version is not peer-reviewed

Submitted:

09 January 2023

Posted:

09 January 2023

You are already at the latest version

Alerts

Abstract

In this paper we study a conjecture proposed by P.Duxbury , C.laror , L.Leduino de Salles Neto in 2021\cite{conjecture} on the Golumb Ruler Problem which is a classical optimization model in discrete case . In \cite{conjecture} the authors constucted a continuous model for the Golumb Ruler Problem associated to the discrete case and conjectured that the optimal value of both cases are equal . We deal with this conjecture via algebraic geometrical methods .

Keywords:

Subject: Computer Science and Mathematics - Applied Mathematics

Contents

1	Introduction	ii

2	The Manin-Brauer Obstruction for $X_{L - 1}$	iv
	2.1 Preliminary.......................................................................................................................................	iv
	2.1.1 A Brief Review of Brauer Group.....................................................................................	iv
	2.1.2 Residue Homomorphism.................................................................................................	v
	2.2 Algebraic-geometry-theoretical Arguments of the Proof............................................................	v
	2.2.1 Step I: Substitution............................................................................................................	vi
	2.2.2 Step II : Local Arguments on $X_{L - 1}$ ..................................................................................	vi

3	Approximating the Real Points	ix
	3.1 Outline of the Proof........................................................................................................................	ix
	3.2 Filling the Gap in the Proof of $X_{L - 1} (R) = \emptyset$ .............................................................................	x
	3.2.1 Shift the Irrational Point to a Rational Point.................................................................	x
	3.2.2 Conclusion........................................................................................................................	xiii

4	Acknowledgement	xiii

1. Introduction

The Golomb Ruler Problem is a classical optimization problem, named for Solomon W. Golomb and discovered independently by Sidon (1932) [25] and Babcock (1953) [1], a Golomb ruler is a finite set of marked points in a ruler at integer positions such that the distances between any two points are different. Furthermore, we say that a Golomb Ruler problem is to find the smallest length of the ruler such that the n marked points form a Golomb ruler. Next, we give the exact definition of a Golomb ruler.

Definition 1.1

(Golomb Rulers). Given a set of positive integers

G = {a_{1}, a_{2}, \dots, a_{n}}

where

a_{1} = 0

and

a_{1} < a_{2} < \dots < a_{n}

, we say that G is a Golomb ruler if and only if for all

i, j, k, l \in {1, 2, \dots, n}

satisfying

a_{i} - a_{j} = a_{k} - a_{l} \Leftrightarrow i = j

and

k = l

. And we call

L_{n} : = a_{n} - a_{0} = a_{n}

the length of the given Golomb ruler.

Given an order n and a length

L_{n}

of the Golomb ruler, the optimal Golomb ruler is referred to the Golomb ruler that is optimally short and exhibits minimal n for the specific value of

L_{n}

Definition 1.2

(Golomb Ruler Problem). Given

n \in Z_{> 0}

. The Golomb ruler problem asks for the minimum length

{\tilde{L}}_{n}

of all Golomb rulers of order n. We call the Golomb ruler with the minimum length for n the optimal Golomb ruler.

For example, for

n = 2

, the optimal Golomb ruler is

{0, 1}

and for

n = 3

the optimal Golomb ruler is

{0, 1, 3}

. One can find more examples of optimal Golomb rulers on Wikipedia. Moreover, Golomb ruler problem is not merely a purely combinatorial problem, it has many profound applications in areas such as astronomy [3], information theory helping to error correcting codes [1,20], radio frequency selection helping to select radio frequencies [2,8]. It is also studied by number theorists like Paul Erdös out of pure mathematical interest [7].

To better understand the Golomb ruler problem, a general way is to reinterpret it as a discrete optimization problem; then we can make use of various powerful tools in mathematical programming [9,26]. One can solve the Golomb ruler problem in a brute-force way, that is, to apply the greedy algorithm to search the optimal Golomb ruler [5]. However, although not proven to be NP-hard, solving the Golomb ruler problem is believed by experts to be very challenging and sophisticated [5,18,24].

In [6] P.Duxbury and his cooperators established a discrete optimization model and a continuous one for the Golomb ruler problem. More surprisingly, they proposed a conjecture which predicated that the optimal value of the discrete model is equal to that of the continuous one. In this paper we study this conjecture utilizing algebraic geometrical techniques and some elementary arguments on rational approximation of real numbers.

In [14], a discrete model for the Golomb ruler problem was first presented but it will not be discussed in this paper. Given a positive integer n and an upper bound

L_{n}

for the length of the ruler, P. Duxbury, C.Lavor, L. Leduino de Salles-Neto [6] modify the form of modeling the Golomb ruler as follows:

min_{x_{i} \in {0, 1}} t

\{\begin{matrix} i x_{i} \leq t, \\ \sum_{i = 1}^{L_{n}} x_{i} = n - 1, \\ x_{j} + \sum_{i = 1}^{L_{n} - j - 1} x_{i} x_{i + j} \leq 1, & j = 1, \dots, L_{n} - 1, \\ x_{i} \in {0, 1}, i = 1, \dots, L_{n} . \end{matrix}

(1)

In [6] a continuous model for the Golomb ruler is also established which can be viewed as a nonlinear relaxation for the above model1:

min_{x_{i} \in [0, 1]} t

\{\begin{matrix} i x_{i} \leq t x_{i}, \\ \sum_{i = 1}^{L_{n}} x_{i} = n - 1, \\ x_{j} + \sum_{i = 1}^{L_{n} - j - 1} x_{i} x_{i + j} \leq 1, & j = 1, \dots, L_{n} - 1, \\ x_{i} \in [0, 1], i = 1, \dots, L_{n} . \end{matrix}

(2)

Moreover, they propose a fascinating conjecture:

Conjecture 1.3

(P. Duxbury, C.Lavor, L. Leduino de Salles-Neto [6]). The optimal value of model (1) is identical to that of model (2).

To illustrate our strategy, it is inevitable to use some algebraic geometry and algebraic number theory terminologies. Our naive insight is to add some non-negative auxiliary parameters to each inequality in 1 to make them equal. Then we can use these equations to define a non-singular algebraic variety

X_{L_{n}}

that contains all the arithmetic and geometric information of 1, where the notation

X (A)

denotes the A-points of an arbitrary algebraic variety

X

and A is some

Q

-algebra. Set L the optimal value of 1, and we directly know that

X_{L - 1} (Z) = \emptyset

. If we further prove that

X_{L - 1} (R) = \emptyset

, then Conjecture 1.3 is implied directly.

We divide the proof of this conjecture into three parts: (1)We regard the feasible regions of 2 and 1 as algebraic schemes after substituting

x_{i} = y_{i}^{2}

and for n and

L_{n}

we write

X_{L_{n}}

to denote the associated schemes; (2) We intend to prove that

X_{L - 1} (Q) = \emptyset

where L is the optimal value for the discrete model 1 ; (3) Lastly we need to verify that

X_{L - 1} (Q)

is dense in

X_{L - 1} (R)

and it follows immediately that

X_{L - 1} (R) = \emptyset

which suffices to prove conjecture 1.3.

The most difficult and technical part turns out to be part (2), in which we utilize algebraic-geometry-theoretical techniques and tools. Given a global field k, the adèle ring of k is defined as the restricted product

A_{k} : = \prod_{v}^{'} (k_{v}, O_{v}) \subseteq \prod_{v} k_{v}

consisting of tuples

(a_{v})

where

a_{v} \in O_{v}

for all but finitely many places v of k, where

O_{v}

is the algebraic integer subring of

k_{v}

. More specifically, it suffices to prove that

X_{L - 1} {(A_{Q})}^{Br} = \emptyset

(we will explain the notation in Definition 2.18) which implies that

X_{L - 1} (Q) = \emptyset

( here

A_{Q}

denotes the adèle ring of

Q

, for a further theory of the adèle ring of a local field, see A. Weil’s textbook [28] or J. Neukirch’s lecture [17] ). Hence we hope to find some Azumaya algebras

{A_{s}}_{s \in S}

indexed by a finite index S such that

⋂_{s \in S} X_{L - 1} {(A_{Q})}^{A_{s}} = \emptyset

(this notation will also be defined in Definition 2.18).

The idea of tackling part (3) is simple. If an

R

-points P exists in

X_{L - 1}

, we can pick out a

Q

-points Q very close to P which also satisfies constraints in 2. This leads to a contraction of the temporarily assumed fact that

X_{L - 1} (Q) = \emptyset

Organization. This paper is organized as follows : In Section 2 we recall some notions and properties in the basic Brauer group theory and realize the goal of Part 2. In Section 3 we finish Part 3.

2. The Manin-Brauer Obstruction for $X_{L - 1}$

The goal of this section is to prove that

X_{L - 1} (Q) = \emptyset

. As mentioned in the Introduction part, the key insight is to find a finite set

S \subseteq Br X_{L - 1}

of Azumaya algebras over

X_{L - 1}

such that

⋂_{A \subseteq S} X_{L - 1} {(A_{Q})}^{A} = \emptyset

2.1. Preliminary

Firstly, let us recall some basic algebraic geometry notions. The main references for this subsection are Hartshorne’s textbook [12] and Poonen’s lecture note [19]. The readers can also check [4,16,23] for more detailed illustrations and advanced topics on algebraic geometry and arithmetic theory. For more detailed and interesting descriptions of Brauer groups and Brauer-Manin obstruction, cf. [21,27], and Grothendieck’s original research on Brauer groups [10,11].

For a precise definition of schemes, the readers can check Section I , II in Chapter 2 in [12].

2.1.1. A Brief Review of Brauer Group

The Brauer group of a field (and more generally a scheme) is an important invariant for studying the arithmetic and geometric information of the field(resp. the scheme). It was originally introduced by Richard Brauer and aims to classify the central simple algebras over a field. And it plays an important role in the construction of Manin-Brauer obstruction.

Definition 2.1

(Azumaya Algebras over Fields). An Azumaya algebra1 over a field F is a F-algebra A (associative and containing the identity element, but not necessarily noncommutative) such that

A \otimes_{F} F^{s e p}

is isomorphic to the matrix algebra

M_{n} (F^{s e p})

as a

F^{s e p}

-algebra for some

n \geq 1

where

F^{s e p}

denotes the separable closure of F. And let

A z_{F}

be the category of Azumaya algebras over F.

Definition 2.2

(Brauer Group of Fields). For a field k, two Azumaya algebras over k are called similar if there exist

m, n \geq 1

such that

M_{n} (A) ≃ M_{m} (B)

as k-algebras. And we write

A \sim B

if they are similar. Then the Brauer group of a field k is defined as follows:

Br k : = A z_{k} / \sim

Definition 2.3.

Let X be a scheme, the (cohomological) Brauer group of X is defined as

Br X : = H_{e t}^{2} (X, G_{m})

However, although the cohomological definition of the Brauer group of schemes is elegant and general, it is not easy to compute and apply explicitly. We will introduce another definition using Azumaya algebra.

Definition 2.4

(Azumaya Algebra over Schemes). An Azumaya algebra on a scheme X is an

O_{X}

-algebra

A

that is coherent as a

O_{X}

-module with

A_{x} \neq 0

for all

x \in X

, that satisfies one of the following conditions:

(i): There is an open covering ${U_{i} \to X}$ in the étale topology such that for each i there exists $r_{i} \in Z_{\geq o}$ such that $A \otimes_{O_{X}} O_{U_{i}} ≅ M_{r_{i}} (O_{U_{i}})$ .
(ii): $A$ is locally free as an $O_{X}$ -module , and the fibre $A (x) : = A \otimes_{O_{X}} k (x)$ is an Azumaya algebra over the residue field $k (x)$ for each $x \in X$ .

Definition 2.5.

Two Azumaya algebra

A

and

A^{'}

on X are similar if there exist locally free coherent

O_{X}

-modules

E

and

E^{'}

of positive rank at each

x \in X

such that

A \otimes_{O_{X}} {End}_{O_{x}} (E) ≃ A^{'} \otimes_{O_{X}} {End}_{O_{x}} (E^{'})

Definition 2.6.

Given a scheme X, the Azumaya Brauer group

{Br}_{Az} X

is the set of similarity classes of Azumaya algebras on X. The multiplication is induced by

A, B \to A \otimes_{O_{X}} B

, the inverse is induced by

A \to A^{opp}

, and the identity is the class of

O_{X}

Fact 1

(Theorem.6.6.17 in [19]). a) For any scheme X , the following map

{Br}_{Az} X \to Br X

is an injective homomorphism.

b): An Azumaya algebra $A$ on X is locally free of rank $n^{2}$ defines an element of ${Br}_{Az} X$ that is killed by n. In particular, if x has at most finitely many connected components, then ${Br}_{Az} X$ is torsion.

2.1.2. Residue Homomorphism

There are two common way to compute the Brauer group. The first is using the Hochschild-Serre spectral sequence in étale cohomology. The second is to apply the residue homomorphism. For practical reason, our proof of rational emptiness is inspired by Iskovskikh’s construction of Mannin-Brauer obstruction to his conic bundle with 4 singular fibers (cf. [13]), we only need to utilize some properties of the residue homomorphism.

Let us recall some important facts for residue homomorphisms .

Proposition 2.7.

Let X be a regular integral notherian scheme . Then the sequence

0 \to Br X \to Br K (X) \overset{r e s}{\to} ⨁_{x \in X^{(1)}} H^{1} (K (X), Q / Z)

is exact, where

K (X)

is the function field of X, with the caveat that the primary p part of all groups must be excluded if X is of dimension

⩽ 1

and some

k (x)

is imperfect of characteristic p, or X is of dimension

⩾ 2

and

k (x)

is of characteristic p.

2.2. Algebraic-geometry-theoretical Arguments of the Proof

In this subsection we explain how to choose a substitution to translate the feasible region of models 1 and 2 and reformulate the conjecture 1.3 in the language of algebraic geometry.

2.2.1. Step I: Substitution

For an integer

n \in Z_{> 0}

, let L be the optimal value of the optimization model 1, and let

x_{i} = \sum_{s = 1}^{4} z_{i, s}^{2}

, where

x_{i}

is with additional restriction

x_{i} \geq 0

, the same as given in the models 2 and 1, for

i = 1, \dots, L - 1

. And we allow

R : = R [z_{1, 1}, z_{2, 1}, z_{3, 1}, z_{4, 1}, \dots z_{1, L - 1}, \dots, z_{4, L - 1}, w_{1, 1}, \dots, w_{4, 1} \dots, w_{1, L - 2}, \dots, w_{4, L - 2}]

X_{L - 1} : = Spec R / (f_{0}, f_{1}, \dots, f_{L - 2})

where the

L - 1

equations are given as:

\{\begin{matrix} f_{0} = \sum_{i = 1}^{L - 1} \sum_{s = 1}^{4} z_{i, s}^{2} - n + 1, \\ f_{j} = \sum_{s = 1}^{4} z_{s, j}^{2} + \sum_{s = 1}^{L - 1 - j} (\sum_{s = 1}^{4} z_{s, i}^{2}) (\sum_{s = 1}^{4} z_{s, i + j}^{2}) + \sum_{s = 1}^{4} w_{s, j}^{2} - 1, j = 1 \dots, L - 2 . \end{matrix}

(3)

As a convention we use

X_{L - 1} (M)

to denote the M-valued points on

X_{L - 1}

where M is any

Q

-algebra. And we observe that:

Observation 2.8.

X_{L - 1} (R) = \emptyset

, then conjeture 1.3 holds.

Proof.

Let

F_{L - 1}

to denote the feasible region determined by the following constraints:

\{\begin{matrix} \sum_{i = 1}^{L - 1} x_{i} = n - 1, \\ x_{j} + \sum_{i = 1}^{L - 1 - j} x_{i} x_{i + j} ⩽ 1 . \\ 0 ⩽ x_{i} ⩽ 1, i = 1, \dots, L - 1 . \end{matrix}

(4)

On the one hand, since L is the optimal value of (1), one can easily verify that 1 and 4 has no integer feasible solutions for

L_{n} = L - 1

. We now give the following explanation: If 4 has an integer solution, denoted as

a = (a_{1}, \dots, a_{L - 1})

, then

(a_{1}, \dots, a_{L - 1}, 0, \dots, 0) \in R^{L_{n}}

is a feasible solution for 1 which implies that the optimal value of 1 is not greater than

L - 1

. This yields a contradiction. Similarly, if 4 has a real solution, then so does 2. The above discussion tells us that to check whether L is the optimal value for the continuous model 2 we only need to know whether 4 has a real solution.

On the other hand, if

X_{L - 1} (R) = \emptyset

which is equivalent to say that the system 3 has no real solution, then 4 is also free of real solutions for the reason that every positive real number can be decomposed as the sum of four squares of real numbers (the same argument also holds for positive rational numbers, i.e. every positive rational number can be decomposed into the sum of four squares of rational numbers) . This implies the conjecture 1.3.

□

The residue part of this section is devoted to proving that the Rational Emptiness of

X_{L - 1}

, that is,

2.2.2. Step II : Local Arguments on $X_{L - 1}$

The following fact inspires us to study the

Q

-points on

X_{L - 1}

Proposition 2.9.

X_{L - 1} (Q) = \emptyset

, then (4) has no rational solutions.

Proof.

This is a straightforward result by Lagrange’s four-square theorem (cf. Landau’s classical textbook [15] for a detailed proof). If there exists a rational solution of (4) , then we can definitely find four rational numbers

z_{1, i}, z_{2, i}, z_{3, i}, z_{4, i} \in Q

such that

x_{i} = \sum_{s = 1}^{4} z_{s, i}^{2}

for each

i \in {1, \dots, L - 1}

, which means that

X_{L - 1} (Q) \neq \emptyset

. This leads to a contradiction.

□

According to the above proposition, we only need to work on

X_{L - 1}

Theorem 2.10.

Let

X_{L - 1}

be as above. Then

X_{L - 1} (Q) = \emptyset

Before proving this theorem, let us recall some basic results of invariant map of Brauer groups.

Lemma 2.11.

Let k be a local field. Then we have the following properties:

(i): There is an injection $inv : Br k \to Q / Z$ , whose image is :

$\{\begin{matrix} \frac{1}{2} Z / Z & if k = R, \\ 0 & if k = C, \\ Q / Z & if k is nonarchimedean . \end{matrix}$

(5)
(ii): Every element of $Br$ k has period equal to index . Especially, if $A \in Br k$ is a quaternion algebra , then ${inv}_{p} A$ is equal to $\frac{1}{2}$ if A is not split or 0 if A is split.

Proof.

One can find the proof of (i) in p.130 of [22] and (ii) in p.25 of [19].

□

Lemma 2.12

(Proposition 1.5.23. in [19]). Given a field k, let

L \supseteq k

be a cyclic extension of k of degree n and let

χ : Gal (\bar{k} / k) \to Z / n Z

be a continuous homomorphism. Then for

a \in k^{\times}

, the k-algebra

(a, χ)

is split if and only if

a \in N_{L / k} (L^{\times})

Proof.

The proof can be found in [19]. □

Remark 2.13.

Notice that a cyclic algebra

(a, χ)

given above is an Azumaya algebra in

Br k

(a, χ)

being split over k means that it is an identity element in

Br k

, hence

{inv}_{v} (a, χ) = 0

for every place v of k.

Lemma 2.14.

Given a field k . And let

a, b, c, d \in k^{\times}

. We have

(a, b) ≃ (a c^{2}, b d^{2})

Proof.

See Exercise 4.(2) in Chapter 2 of [27]. □

Next, we intend to introduce an important notion : the Brauer-Manin obsruction for a k-variety where k is a given field.

Definition 2.15

(Evaluation of the Brauer group). Let

A \in Br X

. If L is a k-algebra and

x \in X (L)

, then

Spec L \overset{x}{\to} X

induces a homomorphism

Br X \to Br L

, which maps A to an element of

Br L

that we call

A (x)

. For more detailed explanations , c.f. Section 8.1.1. in [19].

Now assume that k is a global field and let

A \in Br X

. Then we have:

Proposition 2.16.

(x_{v}) \in X (A_{k})

, then we have

A (x_{v}) = 0

for almost all places v of k.

Proof.

See Proposition 8.2.1 in [19]. □

Proposition 2.17.

x \in X (k) \subseteq X (A_{k})

, then we have

(A, x) = 0

Proof.

See Proposition 8.2.2 in [19]. □

Definition 2.18.

For

A \in Br X

, define that

X {(A_{k})}^{A} : = {(x_{v}) \in X (A_{k}) : (A, (x_{v})) = 0} .

Furthermore we define

X {(A_{k})}^{Br} : = ⋂_{A \in Br X} X {(A_{k})}^{A} .

Proposition 2.19.

It is clear that

X (k) \subseteq X {(A_{k})}^{Br}

Proof.

cf. Corollary 8.2.6 in [19]. □

Before beginning the proof of Theorem2.10 we need to define some necessary quaternion algebras. Let

A_{i j}^{1} : = (1 - z_{i}^{2} z_{i + j}^{2}, - 1)

A_{i j}^{2} : = (1 + z_{i}^{2} z_{i + j}^{2}, - 1)

B_{i j}^{1} : = (\frac{1}{z_{i}^{2}} - z_{i + j}^{2}, - 1)

B_{i j}^{2} : = (\frac{1}{z_{i + j}^{2}} - z_{i}^{2}, - 1)

C_{i j}^{1} : = (\frac{1}{z_{i}^{2}} + z_{i + j}^{2}, - 1)

C_{i j}^{2} : = (\frac{1}{z_{i + j}^{2}} + z_{i}^{2}, - 1)

. It is clear that there is a pair

(i . j)

such that the above quaternion algebras are all in

Br K (X_{L - 1})

according to the assumption that

X_{L - 1} (Z) = \emptyset

We will show that:

Lemma 2.20.

A_{i j}^{δ} = B_{i j}^{δ} = C_{i j}^{δ}

, here

δ \in {0, 1}

and

A_{i j}^{1} = A_{i j}^{2}

. In addition, we have

A_{i j}^{1} \in Br X_{L - 1}

Proof.

The first argument is merely the corollary of Lemma 2.12 and Lemma 2.14. And it is clear that

A_{i j}^{1} \in Br K (X_{L - 1})

. And by Proposition we only need to find out a Zariski open covering

{U_{i}}

X_{L - 1}

such that

A_{i j}^{i}

extends to an element of

Br U_{i}

for each i. Let

P_{1 - z_{i}^{2} z_{i + j}^{2}}

and

P_{1 + z_{i}^{2} z_{i + j}^{2}}

denote the closed points in

P_{Q}^{2}

that

1 - z_{i}^{2} z_{i + j}^{2}

and

1 + z_{i}^{2} z_{i + j}^{2}

vanish, respectively. Then we define

U_{i j}^{1} : = X_{L - 1} -

(fibre above ∞) − (fibre above

P_{1 - z_{i}^{2} z_{i + j}^{2}}

U_{i j}^{2} : = X_{L - 1} -

(fibre above ∞) − (fibre above

P_{1 + z_{i}^{2} z_{i + j}^{2}}

U_{i j}^{3} : = X_{L - 1} -

(fibre above

z_{i + j} = \infty

) − ( fibre above

z_{i} = 0

) − (fibre above

P_{1 - z_{i}^{2} z_{i + j}^{2}}

U_{i j}^{4} : = X_{L - 1} -

(fibre above

z_{i} = \infty

) − ( fibre above

z_{i + j} = 0

) − (fibre above

P_{1 - z_{i}^{2} z_{i + j}^{2}}

U_{i j}^{5} : = X_{L - 1} -

(fibre above

z_{i + j} = \infty

) − ( fibre above

z_{i} = 0

) − (fibre above

P_{1 + z_{i}^{2} z_{i + j}^{2}}

U_{i j}^{6} : = X_{L - 1} -

(fibre above

z_{i} = \infty

) − ( fibre above

z_{i + j} = 0

) − (fibre above

P_{1 + z_{i}^{2} z_{i + j}^{2}}

One immediately sees that

⋃_{i = 1}^{6} U_{i j}^{i} = X_{L - 1}

. And

A_{i j}^{1} \in Br U_{i j}^{1}

A_{i j}^{2} \in Br U_{i j}^{2}

B_{i j}^{1} \in Br U_{i j}^{3}

B_{i j}^{2} \in Br U_{i j}^{4}

C_{i j}^{1} \in Br U_{i j}^{5}

C_{i j}^{2} \in Br U_{i j}^{6}

. Therefore,

A_{i j}^{1}

extends to an element of

Br X_{L - 1}

. □

Lemma 2.21.

In a similar process, we can prove that

D_{i j}^{2} : = (2 - z_{i}^{2} z_{i + j}^{2}, - 1)

D_{i j}^{3} : = (3 - z_{i}^{2} z_{i + j}^{2}, - 1)

D_{i j}^{4} : = (4 - z_{i}^{2} z_{i + j}^{2}, - 1)

are in

Br X_{L - 1}

Proof.

The proof is just repeating the proof of Lemma 2.20. □

Proof of Theorem 2.10

As stated in the Introduction section, we want to find finitely many quaternion algebras to obstruct

X_{L - 1}

. It suffices to show that

⋂_{i, j} (X_{L - 1} {(A_{Q})}^{A_{i j}^{1}} ⋂_{i = 2}^{4} X_{L - 1} {(A_{Q})}^{E_{i j}^{i}}) = \emptyset

Next, we need to calculate

{inv}_{p} A_{i j}^{1} (P)

, and

{inv}_{p} E_{i j}^{k} (P)

, for all places p of

Q

and each

P \in X_{L - 1} (Q_{p})

(1): If $p \notin {2, \infty}$ , we will show that ${inv}_{p} A_{i j}^{1} (P) = 0$ . If $v_{p} (z_{i} z_{i + j}) < 0$ , then $v_{p} (\frac{1}{z_{i}^{2} z_{i + j}^{2}} - 1) = 0$ which means that $\frac{1}{z_{i}^{2} z_{i + j}^{2}} - 1 \in Z_{p}^{\times}$ . If $v_{p} (z_{i} z_{i + j}) \geq 0$ then at least one of $1 - z_{i}^{2} z_{i + j}^{2}$ or $1 + z_{i}^{2} z_{i + j}^{2}$ is in $Z_{p}^{\times}$ . Therefore $A_{i j}^{1} \in Br Z_{p}$ applying the assumption $p \neq 2$ . However, for any local field K, its Brauer group is equal to zero. Hence $Br Z_{p} = 0$ . It follows that ${inv}_{p} A_{i j}^{1} (P) = 0$ when $p \neq 2, \infty$ . The same argument also holds for $E_{i j}^{k}$ , $k = 2, 3, 4$ .
(2): If $p = \infty$ , it is obvious that at least $s + z_{i}^{2} z_{i + j}^{2} > 0$ or $s - z_{i}^{2} z_{i + j}^{2} > 0$ for $s = 1, 2, 3, 4$ , which means that they are all in $N_{Q (\sqrt{- 1}) / R} (R {(\sqrt{- 1})}^{\times}) = R_{> 0}$ . This implies ${inv}_{\infty} A_{i j}^{1} (P) = {inv}_{\infty} D_{i j}^{2} (P) = {inv}_{\infty} D_{i j}^{3} (P) = {inv}_{\infty} D_{i j}^{4} (P) = 0$ .
(3): If $p = 2$ , the discussion is different from that of $p \neq 2$ .

(a) We first study this case when $v_{2} (z_{i} z_{i + j}) > 0$ . One immediately obtains $k - z_{i}^{2} z_{i + j}^{2} \equiv k (mod 4)$ for $k = 1, 2, 3, 4$ . Therefore when $k = 3$ , $3 - z_{i}^{2} z_{i + j}^{2} \equiv - 1 (mod 4)$ which means that $3 - z_{i}^{2} z_{i + j}^{2} \notin N_{Q_{2} (\sqrt{- 1}) / Q_{2}} (Q_{2} {(\sqrt{- 1})}^{\times})$ and ${inv}_{2} D_{i j}^{3} = 1 / 2$ . However, $1 - z_{i}^{2} z_{i + j}^{2} \equiv 1 (mod 4)$ can always be written as a norm form of $N_{Q_{2} (\sqrt{- 1}) / Q_{2}} (Q_{2} {(\sqrt{- 1})}^{\times})$ and it is direct to see that ${inv}_{2} A_{i j}^{1} = 0$ .

(b) If $v_{2} (z_{i} z_{i + j}) = 0$ , then $1 - z_{i}^{2} z_{i + j}^{2}$ , $2 - z_{i}^{2} z_{i + j}^{2}$ , $3 - z_{i}^{2} z_{i + j}^{2}$ , $4 - z_{i}^{2} z_{i + j}^{2}$ also range over the residue class modulo 4, similarly we have exactly one of $A_{i j}^{1}, D_{i j}^{2}, D_{i j}^{3}, D_{i j}^{4},$ whose evaluation at 2 is $1 / 2$ and one of them is 0.

(c) If $v_{2} (z_{i} z_{i + j}) < 0$ , we have $v_{2} (\frac{k}{z_{i}^{2} z_{i + j}^{2}} - 1) = 0$ and then $\frac{k}{z_{i}^{2} z_{i + j}^{2}} - 1 \equiv - 1 (mod 4)$ for all $k = 1, 2, 3, 4$ . That is to say, $\frac{k}{z_{i}^{2} z_{i + j}^{2}} - 1$ is always not a norm form in $N_{Q_{2} (\sqrt{- 1}) / Q_{2}} (Q_{2} {(\sqrt{- 1})}^{\times})$ . Therefore, ${inv}_{2} A_{i j}^{1} = {inv}_{2} D_{i j}^{2} = {inv}_{2} D_{i j}^{3} = {inv}_{2} D_{i j}^{4} = 1 / 2$ .

Combining the computation results from (1), (2) and (3), we achieve that

⋂_{i, j} (X_{L - 1} {(A_{Q})}^{A_{i j}^{1}} ⋂_{i = 2}^{4} X_{L - 1} {(A_{Q})}^{E_{i j}^{i}}) = \emptyset

□

3. Approximating the Real Points

3.1. Outline of the Proof

The proof of the emptiness of real-valued points depends only on the original form of 1.3 and will not involve any algebraic-geometry-theoretical notions and techniques. If a real point exists in

X_{L - 1}

, it is equivalent to claim that 2 has a real-valued feasible point B. Hence our strategy is to return to model 2 and find a feasible rational point close enough to B that also lies in the feasible region of 2. This will contradict with

X_{L - 1} (Q) = \emptyset

we have proven in Section 2.

We let

x = (x_{1}, \dots, x_{L - 1})

g_{j} (x) = x_{j} + \sum_{i = 1}^{L - 1 - j} x_{i} x_{i + j}

for

j = 1, \dots, L - 2

. And explicitly we write

B = (t_{1}, \dots, t_{L - 1}) \in R^{L - 1}

which is the hypothetical real-valued feasible solution of the following equations as the same as given in (4).

\{\begin{matrix} \sum_{i = 1}^{L - 1} x_{i} = n - 1, \\ g_{j} (x) \leq 1, f o r j = 1, \dots, L - 2 . \\ 0 ⩽ x_{i} ⩽ 1, i = 1, \dots, L - 1 . \end{matrix}

(6)

Notice 3.1.

In this section we use the notation

(a, b)

to denote an open interval in the real axis where

a, b \in R

. And don’t be confused with the notation of a quaternion algebra in Section 2.

The main objective of this section is to prove the following:

Theorem 3.2.

(6) does not have a real feasible solution.

Sketch of the Proof

Let B be as above. The special case is that all

g_{j} (B) < 1

. We firstly work in this case. Suppose

ϵ = \frac{{min}_{1 \leq j \leq L - 2} (1 - g_{j} (B))}{2 (L - 1)} .

Define a neighborhood of B as

U_{ϵ} : = \prod_{i = 1}^{L - 1} (t_{i} - ϵ, t_{i} + ϵ)

. It is easy to check that the intersection of

U_{ϵ}

and the hyperplane

P_{0} : \sum_{i = 1}^{L - 1} x_{i} = n - 1

is lie in the feasible region of (6). However, by the density of

Q^{L - 1} \subseteq R^{L - 1}

, one can definitely find one rational point in

U_{ϵ} ⋂ P_{0}

and we will demonstrate why this fact is true in Lemma 3.3. This leads to a contradiction with

X_{L - 1} (Q) = \emptyset

In the general cases, let

I \subseteq {1, 2, \dots, L - 2}

to denote the index set such that

g_{n} (B) = 1

for each

n \in I

. And write

\bar{I}

the complement of I in

{1, \dots, L - 2}

.That is,

\{\begin{matrix} g_{n} (B) = 1, i f n \in I, \end{matrix} g_{n} (B) < 1, i f n \in \bar{I} .

(7)

And suppose

δ : = {min}_{j \in \bar{I}} (1 - g_{j} (B)) / 2 (L - 1)

. Define an open subset

U_{δ} : = \prod_{i \in I} (t_{i} - δ, t_{i}) \times \prod_{i \in \bar{I}} (t_{i} - δ, t_{i} + δ)

. One can also find a rational point in

U_{δ} ⋂ P_{0}

which is also open in

R^{L - 1}

. This also contradicts the fact that

X_{L - 1} (Q) = \emptyset

(we will give a proof of this issue in Lemma 3.3 in the next subsection.

□

3.2. Filling the Gap in the Proof of $X_{L - 1} (R) = \emptyset$

3.2.1. Shift the Irrational Point to a Rational Point

In this subsection, we prove some technical issues for the front subsection. The main goal is to construct a rational feasible point from the given hypothetical point B.

Lemma 3.3.

Let

U_{ϵ}

and

P_{0}

be the same as defined in the proof of Theorem 3.2. Then (1) when

g_{j} (x) < 1

for all

1 \leq j \leq L - 1

U_{ϵ} ⋂ P_{0} ⋂ Q^{L - 1}

is nonempty and contained in the feasible region of the system (6); (2) generally, let δ,

U_{δ}

be as stated in the last paragraph in the previous subsection,

U_{δ} ⋂ P_{0} ⋂ Q^{L - 1}

is also nonempty.

Proof. (1) Firstly, we need to show that

U_{ϵ}

lies in the feasible region of

g_{j} (x) < 1

. Let

u \in U_{ϵ}

. Then

\begin{matrix} g_{j} (u) - g_{j} (B) & = u_{j} + \sum_{i = 1}^{L - 1 - j} u_{i} u_{i + j} - t_{j} - \sum_{i = 1}^{L - 1 - j} t_{i} t_{i + j} \end{matrix}

(8)

\begin{matrix} ⩽ ϵ + \sum_{i = 1}^{L - 1 - j} (2 ϵ + ϵ^{2}) \end{matrix}

(9)

\begin{matrix} < 2 (L - 1) ϵ \end{matrix}

(10)

\begin{matrix} = min_{1 \leq j \leq L - 2} (1 - g_{j} (B)) \end{matrix}

(11)

Hence

g_{j} (u) ⩽ 1

for all

1 ⩽ j ⩽ L - 2

. The above arguments justify that

U_{ϵ}

lies in the feasible region of

g_{j} (x) < 1

. Then it remains to show that

U_{ϵ} ⋂ P_{0} ⋂ Q^{L - 1}

is nonempty.

It is easy to construct such a rational point in

U_{ϵ} ⋂ P_{0} ⋂ Q^{L - 1}

. Not uniquely, choose

s_{1} \in (t_{1} - \frac{ϵ}{L - 1}, t_{1} + \frac{ϵ}{L - 1}) ⋂ Q

i = 1, \dots, L - 2

, and choose

s_{L - 1} = n - 1 - \sum_{i = 1}^{L - 2} s_{i}

. It is clear that

\begin{matrix} | s_{L - 1} - t_{L - 1} | & ⩽ \sum_{i = 1}^{L - 2} | s_{i} - t_{i} | \end{matrix}

(12)

\begin{matrix} < ϵ \end{matrix}

(13)

It implies that

(s_{1}, \dots, s_{L - 1}) \in U_{ϵ} ⋂ P_{0} ⋂ Q^{L - 1}

(2) The proof of the general part is similar but requires a more delicate shift of each

x_{i}

. Set

{α_{1}, α_{2}, \dots, α_{t}}

an irrational basis of

{t_{1}, \dots, t_{L - 1}}

where

α_{1} = 1

, here by an irrational basis we mean that each

t_{i}

can be uniquely written as the summation from

α_{1}

α_{t}

with rational coefficients. If

t = 1

, then all terms

t_{i}

are rational numbers that contradict

X_{L - 1} (Q) = \emptyset

. We focus on the case where

t > 2

. Write

t_{i} = \sum_{p = 1}^{t} a_{i, p} α_{p}, i = 1, \dots, L - 1 .

For some computational reasons, we intend to properly choose the basis for each coefficient

a_{i, p} ⩾ 0

. Assume

1 = α_{1} ⩽ α_{2} ⩽ \dots ⩽ α_{t}

, and set

ρ_{p} : = {min}_{1 ⩽ i ⩽ L - 1} a_{i, p}

for each

p \in {1, \dots, t}

. And suppose that

α_{t}^{'} : = θ α_{t} + \sum_{p = 1}^{t - 1} ρ_{p} α_{p}

, where

θ

is the minimum integer such that

θ α_{t} + \sum_{p = 1}^{t - 1} ρ_{p} α_{p} > 0

. It is clear that

{α_{1}, \dots, α_{t - 1}, α_{t}^{'}}

is also a transcendental basis for

{t_{1}, \dots, t_{L - 1}}

. Furthermore, under this basis

{α_{1}, \dots, α_{t - 1}, α_{t}^{'}}

, the coefficient

a_{i, p}^{'}

α_{p}

for

t_{i}

is always non-negative. We also use this basis in the following computations.

And we have

\begin{matrix} \sum_{i = 1}^{L - 1} t_{i} & = \sum_{i = 1}^{L - 1} \sum_{p = 1}^{t} a_{i, p}^{'} α_{p}^{'} \end{matrix}

(14)

\begin{matrix} = \sum_{p = 1}^{t} c_{p} α_{p}^{'} \end{matrix}

(15)

\begin{matrix} = n - 1, \end{matrix}

(16)

where

c_{p} = \sum_{i = 1}^{L - 1} a_{i, p}^{'}

. It implies that

c_{1} = n - 1

and

c_{p} = 0

for all

p ⩾ 2

for

{α_{1}, \dots, α_{t}^{'}}

are linearly independent over

Q

. Moreover, rewriting other equations (

g_{j} (B) = 1

when

j \in I

) under this basis we have

\begin{matrix} x_{j} + \sum_{i = 1}^{L - 1 - j} x_{i} x_{i + j} & = \sum_{p = 1}^{t} a_{i, p}^{'} α_{p} + \sum_{i = 1}^{L - 1 - j} (\sum_{p = 1}^{t} a_{i, p}^{'} α_{p}) \cdot (\sum_{p = 1}^{t} a_{i + j, p}^{'} α_{p}) \end{matrix}

(17)

\begin{matrix} = \sum_{i = 1}^{p} d_{i, p} α_{p}^{'} + \sum_{\begin{matrix} 2 ⩽ p ⩽ t \\ 2 ⩽ q ⩽ t \end{matrix}} l_{j p q} α_{p}^{'} α_{q}^{'} \end{matrix}

(18)

\begin{matrix} = 1, j \in I \end{matrix}

(19)

where

d_{i, p}

is the sum of rational coefficients in (17) while

l_{j p q}

is the sum of coefficients of

α_{p} α_{q}

. Note that the coefficients

d_{i, p}

and

l_{j p q}

in (18) are all nonnegative. If we substitute

α_{p}^{'}

for a slightly smaller rational number, the value of (15) is still equal to

n - 1

and the value of (18) will not exceed 1. Choose t positive rational numbers

{π_{p}}_{1 ⩽ p ⩽ t}

where

π_{p}

is in the interval

(max \{0, α_{p} - \frac{δ}{max {t, t \cdot {max}_{i, p} | a_{i, p}^{'} |}}\}, α_{p}),

and

δ = {min}_{j \in \bar{I}} (1 - g_{j} (B)) / 2 (L - 1)

is as defined in the previous subsection. Then we substitute all

α_{p}^{'}

with

π_{p}

and define

b_{i} : = \sum_{p = 1}^{t} a_{i, p}^{'} π_{p}, i = 1, \dots, L - 1 .

Of course we have

0 ⩽ b_{i} ⩽ t_{i} ⩽ 1

. Moreover, we have

\begin{matrix} | b_{i} - t_{i} | & ⩽ \sum_{p = 1}^{t} a_{i, p}^{'} | π_{p} - α_{p}^{'} | \end{matrix}

(20)

\begin{matrix} < \sum_{p = 1}^{t} δ / t \end{matrix}

(21)

\begin{matrix} = δ . \end{matrix}

(22)

Without a doubt,

b_{i}

is in

Q

for all

1 ⩽ i ⩽ L - 1

. Denote

Q = (b_{1}, \dots, b_{L - 1}) \in Q^{L - 1}

. We also have:

\begin{matrix} \{\begin{matrix} \sum_{i = 1}^{L - 1} b_{i} = n - 1, \\ g_{j} (Q) = b_{j} + \sum_{i = 1}^{L - 1 - j} b_{i} b_{i + j} ⩽ 1, j \in I . \end{matrix} \end{matrix}

(23)

And for

j \in \bar{I}

\begin{matrix} | g_{j} (Q) - g_{j} (B) | & = | (b_{j} - t_{j}) + \sum_{i = 1}^{L - 1 - j} (b_{i} b_{i + j} - t_{i} t_{i + j}) | \end{matrix}

(24)

\begin{matrix} < δ + \sum_{i = 1}^{L - 1 - j} ((t_{i} + t_{i + j}) δ + δ^{2}) \end{matrix}

(25)

\begin{matrix} < 2 (L - 1 - j) δ + δ + (L - 1 - j) δ^{2} \end{matrix}

(26)

\begin{matrix} ⩽ 2 (L - j) δ \end{matrix}

(27)

\begin{matrix} < min_{j \in \bar{I}} (1 - g_{j} (B)) . \end{matrix}

(28)

It follows that

g_{j} (Q) ⩽ 1

for all

1 ⩽ j ⩽ L - 1

. That is to say, Q lies in the the feasible region of 6 as a rational point.

□

3.2.2. Conclusion

Combining results in Section 2 and Section 3 we give an entire proof of Conjecture 1.3. The method we use in the proof is purely algebraic and arithmetic including some elementary computations. In general, one can expect to reformulate an optimization problem as an algebraic geometric problem which is extremely powerful to deal with some special problems, for examples, the existence of integer feasible solutions of a given optimization model whose constraints are all rational polynomials. We also expect to find more optimization models whose discrete case and continuous case have the same optimal value.

Acknowledgments

First, the first author, L.Tianyu, sincerely thanks his supervisor LUO CUICUI (last author of the article) for her patient guidance and encouragements during the period in writing this article, sincerely. Lastly, the authors sincerely thank all the referees. This work was supported by Technology and Innovation Major Project of the Ministry of Science and Technology of China under Grant 2020AAA0108400, 2020AAA0108402 and 2020AAA0108404.

Conflicts of Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

References

Babcock, W.C. Intermodulation interference in radio systems frequency of occurrence and control by channel selection. The Bell System Technical Journal 1953, 32, 63–73. [Google Scholar] [CrossRef]
Babcock, W.C. Intermodulation interference in radio systems frequency of occurrence and control by channel selection. The Bell System Technical Journal 1953, 32, 63–73. [Google Scholar] [CrossRef]
Blum, E.; Ribes, J.; Biraud, F. Some new possibilities of optimum synthetic linear arrays for radioastronomy. Astronomy and Astrophysics 1975, 41, 409–411. [Google Scholar]
Conrad, B.D. Arithmetic algebraic geometry, Volume 9. American Mathematical Soc., 2001.
Dollas, A.; Rankin, W.T.; McCracken, D. A new algorithm for Golomb ruler derivation and proof of the 19 mark ruler. IEEE Transactions on Information Theory 1998, 44, 379–382. [Google Scholar] [CrossRef]
Duxbury, P.; Lavor, C.; de Salles-Neto, L.l. A conjecture on a continuous optimization model for the Golumb Ruler Problem. RAIRO–Operations Research.
Erdos, P.; Turán, P. On a problem of Sidon in additive number theory, and on some related problems. J. London Math. Soc 1941, 16, 212–215. [Google Scholar] [CrossRef]
Fang, R.J.; Sandrin, W.A. Carrier frequency assignment for nonlinear repeaters. COMSAT Technical Review 1977, 7, 227–245. [Google Scholar]
Galinier, P.; Centre for Research on Transportation (Montréal, Q. A constraint-based approach to the Golomb ruler problem; Université de Montréal, Centre de recherche sur les transports, 2003.
Grothendieck, A. Le groupe de Brauer. I. Algèbres d’Azumaya et interprétations diverses. Dix exposés sur la cohomologie des schémas 1968, 3, 15. [Google Scholar]
Grothendieck, A. Le groupe de Brauer. II. Théorie cohomologique. Dix exposés sur la cohomologie des schémas 1968, 3, 67–87. [Google Scholar]
Hartshorne, R. Algebraic geometry; Volume 52, Springer Science & Business Media, 2013.
Iskovskikh, V.A. Minimal models of rational surfaces over arbitrary fields. Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya 1979, 43, 19–43. [Google Scholar]
Kocuk, B.; Hoeve, W.J.v. A computational comparision of optimization methods for the Golumb ruler problem. In International Conference on Intergration of Constraint Programming, Artificial Intelligence and Operation Research. Springer, 2019, pp. 409–425.
Landau, E. Elementary number theory; Volume 125, American Mathematical Society, 2021.
Liu, Q.; Erne, R. Algebraic Geometry and Arithmetic Curves; Volume 6, Oxford University Press, 2006.
Neukirch, J. Algebraic number theory; Volume 322, Springer Science & Business Media, 2013.
Oshiga, O.; Abreu, G. Design of orthogonal Golomb rulers with applications in wireless localization. In proceedings of the 2014 48th Asilomar Conference on Signals, Systems and Computers. IEEE, 2014, pp. 1497–1501.
Poonen, B. Rational points on varieties; Volume 186, American Mathematical Soc., 2017.
Robinson, J.; Bernstein, A. A class of binary recurrent codes with limited error propagation. IEEE Transactions on Information Theory 1967, 13, 106–113. [Google Scholar] [CrossRef]
Scharaschkin, V. Local-global problems and the Brauer-Manin obstruction; University of Michigan, 1999.
Serre, J.P. Local Fields; Vol. 67, Springer Science & Business Media, 2013.
Shafarevich, I.R.; Reid, M. Basic algebraic geometry; Volume 2, Springer, 1994.
Shearer, J.B. Some new optimum Golomb rulers. IEEE Transactions on Information Theory 1990, 36, 183–184. [Google Scholar] [CrossRef]
Sidon, S. Ein Satz über trigonometrische Polynome und seine Anwendung in der Theorie der Fourier-Reihen. Mathematische Annalen 1932, 106, 536–539. [Google Scholar] [CrossRef]
Smith, B.M.; Stergiou, K.; Walsh, T. Modelling the Golomb ruler problem. RESEARCH REPORT SERIES-UNIVERSITY OF LEEDS SCHOOL OF COMPUTER STUDIES LU SCS RR.
Voight, J. Quaternion algebras; Springer Nature, 2021.
Weil, A. Basic number theory.; Volume 144, Springer Science & Business Media, 2013.

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

Copyright: This open access article is published under a Creative Commons CC BY 4.0 license, which permit the free download, distribution, and reuse, provided that the author and preprint are cited in any reuse.

MDPI Initiatives

Important Links

Choose an area of interest and we will send you notifications of new preprints at your preferred frequency.

Disclaimer

The Proof of a Conjecture For a Continuos Golumb Ruler Model

Abstract

1. Introduction

2. The Manin-Brauer Obstruction for X L − 1

2.1. Preliminary

2.1.1. A Brief Review of Brauer Group

2.1.2. Residue Homomorphism

2.2. Algebraic-geometry-theoretical Arguments of the Proof

2.2.1. Step I: Substitution

2.2.2. Step II : Local Arguments on X L − 1

3. Approximating the Real Points

3.1. Outline of the Proof

3.2. Filling the Gap in the Proof of X L − 1 ( R ) = ∅

3.2.1. Shift the Irrational Point to a Rational Point

3.2.2. Conclusion

Acknowledgments

Conflicts of Interest

References

MDPI Initiatives

Important Links

Subscribe

2. The Manin-Brauer Obstruction for $X_{L - 1}$

2.2.2. Step II : Local Arguments on $X_{L - 1}$

3.2. Filling the Gap in the Proof of $X_{L - 1} (R) = \emptyset$