Algorithmic aspects of arithmetical structures
aa r X i v : . [ m a t h . N T ] J a n ALGORITHMIC ASPECTS OF ARITHMETICAL STRUCTURES
CARLOS E. VALENCIA AND RALIHE R. VILLAGR ´AN
Abstract.
Arithmetical structures on graphs were first mentioned in [10] by D. Lorenzini. Later in[3] they were further studied on square non-negative integer matrices. In both cases, necessary andsufficient conditions for the finiteness of the set of arithmetical structures were given. Therefore, itis natural to ask for an algorithm that compute them.This article is divided in two parts. In the first part we present an algorithm that computesarithmetical structures on a square integer non-negative matrix L with zero diagonal. In order todo this we introduce a new class of Z-matrices, which we call quasi M -matrices. We recall thatarithmetical structures on a matrix L are solutions of the polynomial Diophantine equation f L ( X ) := det(Diag( X ) − L ) = 0 . In the second part, the ideas developed to solve the problem over matrices are generalized to a widerclass of polynomials, which we call dominated. In particular the concept of arithmetical structureis generalized on this new setting. All this leads to an algorithm that computes arithmeticalstructures of dominated polynomials. Moreover, we show that any other integer solution of adominated polynomial is bounded by a finite set and we explore further methods to obtain them.
Keywords:
Arithmetical structures, Diophantine equation, M -matrix, Hilbert’s tenth problem. AMS Mathematical Subject Classification 2020:
Primary 11D72,11Y50; Secondary 11C20,15B48 Introduction
Given a non-negative integer matrix L with zero diagonal, a pair ( d , r ) ∈ N n + × N n + is called anarithmetical structure of L if(1.1) (Diag( d ) − L ) r t = t and gcd( r , . . . , r n ) = 1 . We impose the condition of primitiveness, gcd( r , . . . , r n ) = 1, on vector r because (Diag( d ) − L ) r t = t implies that (Diag( d ) − L ) c r t = t for all c ∈ N + . The set of arithmetical structures on L isdenoted by A ( L ).Arithmetical structures were first introduced for graphs, more precisely when L is the adjacencymatrix of a graph, by D. Lorenzini in [10] as some intersection matrices that arise in the study ofdegenerating curves in algebraic geometry. A more combinatorial aspect of arithmetical structureson graphs have been studied in [2], [4] and [16]. Unless otherwise specified, L will always denote asquare integer non-negative matrix of size n with zero diagonal.For simplicity, when L is the adjacency matrix A ( G ) of a graph G , its set of arithmetical structuresis denoted by A ( G ) instead of A ( A ( G )). Also, we indistinctly say that ( d , r ) ∈ A ( G ) is either anarithmetical structure of the graph G or its incidence matrix A ( G ). The pseudolaplacian matrix of G and d is the matrix L ( G, d ) = Diag( d ) − A ( G ) . When d = deg G is the degree vector of G , then L ( G, deg G ) is its Laplacian matrix.It is important to recall that the set of arithmetical structures on a simple connected graph is finite,see Theorem 1.1. This result was generalized to non-negative matrices in [3]. Before presenting Carlos E. Valencia was partially supported by SNI and Ralihe R. Villagr´an by CONACyT. this result, let us recall that a matrix A is called reducible whenever there exists a permutationmatrix P such that: P t AP = (cid:18) X Y Z (cid:19) . That is, A is similar via a permutation to a block upper triangular matrix. We say that A isirreducible when is not reducible. Equivalently, when A is the adjacency matrix of a digraph, A is irreducible if and only if the digraph associated to A is strongly connected, see [6]. Now we areready to present the finiteness result. Theorem 1.1. [3, Theorem 3.8] If L is an integer non-negative matrix with zero diagonal, then A ( L ) is finite if and only if L is irreducible. Remark 1.2.
When L is a block matrix its arithmetical structures can be obtained from the arith-metical structures of its diagonal blocks. Something similar when L is reducible. Since the set of arithmetical structures is finite, it is natural to ask:
Question 1.3.
Is there an algorithm that computes arithmetical structures of an integer non-negative matrix with zero diagonal?
It is easy to check that every vector d of an arithmetical structure ( d , r ) of L is a solution of thepolynomial Diophantine equation f L ( X ) := det(Diag( X ) − L ) = 0 . Caution 1.4.
The converse is false, see Example 3.1.
Therefore computing arithmetical structures of a matrix consist on computing a subset of thesolutions of a very special class of Diophantine equations, those whose polynomial is the determinantof a matrix with variables in the diagonal.The tenth problem on Hilbert’s list asks to find an algorithm such that given any polynomialDiophantine equation it determines whether it has a solution over the integers. Let R be a ring,then Hilbert’s tenth problem for R (HTP( R )) is to determine if there is an algorithm such that itcan classify any polynomial to whether have a solution in R or not. Based on important preliminarywork by Martin Davis, Hilary Putnam and Julia Robinson, Yuri V. Matiyasevich showed in 1970that HTP( Z ) has a negative answer. That is, that in general no such algorithm exists (see [11]).Since then, several authors have proposed different versions of HTP. Now, if F is a set of polynomialswith coefficients on R , we state the following problem:Is there exists an algorithm that say us when a polynomial p ∈ F has a solution in R ornot.Let us call this the Hilbert’s tenth problem for F , denoted by HTP( F , R ). Note that if F consistsof all polynomials with coefficients in R , then HTP( F , R ) is equivalent to HTP( R ).Before Matiyasevich’s negative solution for Z there were some efforts to build algorithms for somespecial families of polynomial Diophantine equations. In this article we reclaim that approach fora special set of polynomials. Moreover, we find that any other integer solution of the Diophantineequation f L ( X ) = 0 is bounded by a finite set of vectors, which contains as a subset the arithmeticalstructures of the matrix. That is, in this very special case HTP can be solved in a positive way(HTP( { polynomials of the form f L ( X ) } , Z )).At this point we must also ask: Question 1.5.
What does make this type of Diophantine equation so special in such a way that wecan compute all of its positive integer solutions?
LGORITHMIC ASPECTS OF ARITHMETICAL STRUCTURES 3
We will answer by introducing the class of polynomials f that have a monomial m such that anyother monomial of f is a factor of m . We call them dominated polynomials, see Definition 4.1.In figure 1 we illustrate where this work is placed in the framework of HTP. Recall that any ring R such that Z ( R ( Q is of the form R = Z [ S − ], where S is a subset of the prime numbers P . HTP
HTP( Q ) ≡ HTP( Z [ S − ] )(for some S infinite and co-infinite and for any S co-finite); rings of integers of number fields U n s o l v e d Thue-Mahler equations;HTP( { elliptic curves } , Z );HTP( D I , Z ) (see section 5) fin i t e l y m a n y s o l u t i o n s Pell’s equations,for instance infinitelymany solutions see [15] forseveral examples n o e x p li c i t s o l u t i o n s T h e r e i s a n a l go r i t h m / m e t h o d t h a t find s ... HTP( Z ), HTP( Z [ S − ] )for any S finite, andsome S inf. and co-inf. T h e r e i s n o a l g o r i t h m S o l v e d Figure 1.
Matiyasevich gives a negative answer to HTP( Z ). In [13] it was proventhat if S is finite, then HTP( Z [ S − ]) has a negative answer. In fact, by [12] thereare some infinite but co-infinite ( P − S infinite) sets S such that HTP( Z [ S − ])has a negative answer as well. We refer the reader to [15] for different algorithmicapproaches to HTP for several families of Diophantine equations. In this paper wesolve HTP for D I (irreducible dominated polynomials) over Z and an algorithm thatcomputes all arithmetical structures explicitly is given. On the other hand, HTP( Q )is still opened and if P − S is finite then HTP( Z [ S − ]) is equivalent to HTP( Q ) by[13]. In [5] several examples are constructed with S both infinite and co-infiniteand such that HTP( Z [ S − ]) is equivalent to HTP( Q ). HTP for rings of integers ofnumber fields remains open in general.The purpose of this article is to answer questions 1.3 and 1.5. Thus we divide it into two parts:In the first part we give an algorithm that computes the arithmetical structures of an integer non-negative matrix with zero diagonal and in the second part we generalize our result to the new classof multivariate dominated polynomials. This first part consists on sections 2 and 3. In section 2,we recall some theory about M -matrices as given in [3]. Additionally we introduce the class ofquasi M -matrices. Which has all their proper principal minors are positive, but unlike M -matricesand almost nonsingular M -matrix its determinant is not necessarily non-negative. Moreover, wewill establish some properties of these matrices that help us to find the algorithm and prove itscorrectness.Now, let D ≥ ( L ) = { d ∈ N n + | (Diag( d ) − L ) is an almost non-singular M -matrix } , where L is a square integer non-negative matrix L with zero diagonal. By Dickson’s Lemma theset of minimal elements min D ≥ ( L ) of D ≥ ( L ) is finite. In Section 3, we present an algorithm CARLOS E. VALENCIA AND RALIHE R. VILLAGR ´AN that computes min D ≥ ( L ), see Algorithm 3.3. Using this algorithm as a subrutine we get a secondalgorithm that computes the arithmetical structures on L , see Algorithm 3.5. At the end of thissection we use the algorithm developed to present some computational evidence for the followingconjecture. Conjecture 1.6. [3, Conjecture 6.10]
Let G be a simple graph with n vertices, then (cid:12)(cid:12)(cid:12) A ( P n ) (cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12) A ( G ) (cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12) A ( K n ) (cid:12)(cid:12)(cid:12) , where P n and K n are the path and the complete graph on n vertices respectively. The second part of this article consists on sections 4 and 5. In Section 4, we introduce the conceptof arithmetical structures for a general integer square-free dominated polynomial and study some ofits algorithmic aspects. More precisely, we give an algorithm that computes some minimal elementsfor this wider class of polynomials, see Algorithm 4.7. We give an example of a polynomial that isnot the determinant of an integer matrix and how Algorithm 4.7 works for it. Lastly we explorethe limits of the algorithm with several examples.Throughout this article we use the usual partial order over R n given by a ≤ b if and only if a i ≤ b i for all i = 1 , . . . , n and a , b ∈ R n . In a similar way, a < b if and only if a ≤ b and a = b . It is wellknown that this is a well partial order over N n . The following property of subsets of N n under thispartial order is known as Dickson’s Lemma. Lemma 1.7. [9]
For any S ⊆ N n , the set min( S ) = { x ∈ S | y (cid:2) x ∀ y ∈ S } of minimal elements of S under the usual partial order ≤ is finite. Arithmetical Structures on non-negative matrices
In this section we recall the concepts of arithmetical structures and M -matrices. Additionally weintroduce a wider class of matrices that share some properties with M -matrices. This new class ofmatrices, called quasi M -matrices, will be useful in the construction of the algorithm that computesthe arithmetical structures on a matrix. Henceforth we assume that by “matrix” we mean a squarematrix of size n × n for some positive integer n unless the contrary is stated.Given an integer non-negative matrix L with diagonal zero, let A ( L ) = n ( d , r ) ∈ N n + × N n + (cid:12)(cid:12)(cid:12) (Diag( d ) − L ) r t = t and gcd { r i | i = 1 , . . . , n } = 1 o be the set of arithmetical structures on L . Also, let D ( L ) = (cid:8) d ∈ N n + (cid:12)(cid:12) ( d , r ) ∈ A ( L ) (cid:9) and R ( L ) = (cid:8) r ∈ N n + (cid:12)(cid:12) ( d , r ) ∈ A ( L ) (cid:9) , be the sets of d-arithmetical structures and r-arithmetical structures of L respectively. As the nextresult shows it is not difficult to characterize when A ( L ) is non empty. Proposition 2.1. If L is a non-negative matrix with zero diagonal, then A ( L ) = ∅ if and only if L has no row with all entries equal to zero.Proof. ( ⇒ ) If A ( L ) = ∅ , then there exists d , r ∈ N n + such that [diag( d ) − L ] r t = t . Thus( L r t ) i = d i r i ≥ ≤ i ≤ n . Moreover, since L is integer non-negative and r ≥
1, then L t ≥ t . That is, L has no row with all entries equal to zero.( ⇐ ) Since L is integer non-negative and has no row with all entries equal to zero, then L t ≥ t .Thus (Diag( L t ) − L ) t = t and therefore ( L t , ) is an arithmetical structure of L . (cid:3) LGORITHMIC ASPECTS OF ARITHMETICAL STRUCTURES 5
When L t ≥ t (that is, L has all its rows different to ), the arithmetical struture ( L t , ) iscalled the canonical (or trivial) arithmetical structure of L . In a similar way, the next result givesus a necessary and sufficient condition for the finiteness of A ( L ). Theorem 2.2. [3, Theorem 3.8]
Let L be a non-negative matrix with zero diagonal such that A ( L ) = ∅ . Then A ( L ) is finite if and only if L is irreducible. Remark 2.3.
Note that if L is irreducible, then − L and ( D − L ) are irreducible as well, forany diagonal matrix D > . Also note that this theorem means that a multidigraph (with positiveoutdegree on every vertex) has a finite number of arithmetical structures if and only if it is stronglyconnected. Example 2.4.
Using Proposition 2.1 it is not difficult to check that if a digraph D has a vertexwith outdegree zero, then A ( A ( D )) = ∅ . On the other hand, given any strongly connected digraph D = ( V, E ) , let D x be the digraph given by V ( D x ) = V ( D ) ∪ { x } with x / ∈ V ( D ) and E ( D x ) = E ( D ) ∪ { ( x, y ) | y ∈ V ( D ) } . The digraph D x has a vertex with indegree equal to zero and is not strongly connected, therefore ithas an infinite number of arithmetical structures. Note that the adjacency matrix of D x is indeedreducible, A ( D x ) = (cid:18) A ( D ) (cid:19) . However, the multidigraph resulting from D x by reverting the orientation of its arcs has nonearithmetical structures. M -matrices. A real matrix M = ( m ij ) is called a Z-matrix if m ij ≤ i = j . M -matrices can be defined in several ways, see [1]. In this paper we focus on the following definition.Z-matrix Definition 2.5. [1, Theorem 6.4.6 ( A ), page 156] A Z-matrix A is called an M -matrix if all ofits principal minors are non-negative. Furthermore an M -matrix is non-singular if and only if all of its principal minors are positive.Non-singular and singular matrices were studied in [1, Chapter 6]. M -matrices are present in alarge variety of mathematical subjects, like numerical analysis, probability, economics, operationsresearch, etc., see [1] and the references therein. The next class of M -matrices were introducedin [3]. Definition 2.6. An M -matrix M is an almost non-singular M -matrix if all of its proper principalminors are positive. Thus, M is an almost non-singular M -matrix of size n if and only if all of its proper sub-matricesof size n − M -matrices and det( M ) ≥
0. The next result relates arithmeticalstructures on a matrix and M -matrices. Theorem 2.7. [3, Theorem 3.2]
Let M be a Z-matrix. If there exists r with all its entries positivesuch that M r t = t , then M is an M -matrix. Moreover, M is an almost non-singular M -matrixwith det( M ) = 0 if and only if M is irreducible and there is a vector r > such that M r t = t . Therefore when M is an irreducible Z-matrix, the concept of arithmetical structure is equivalent tothat of almost non-singular M -matrix. A direct consequence of Theorem 2.7 is the following result. Corollary 2.8. [3, Corollary 3.3] If M is an irreducible Z-matrix, then there exists r with all itsentries positive such that M r t = 0 if and only if there exists s with all its entries positive such that M t s t = 0 . CARLOS E. VALENCIA AND RALIHE R. VILLAGR ´AN
Thus Corollary 2.8 implies that if L is a non-negative matrix with zero diagonal L , then L and L t have the same set of d -arithmetical structures, but not necessarily the same set of r -arithmeticalstructures. Now, let us present the next properties of almost non-singular M -matrices. Theorem 2.9. [3, Theorem 2.6] If M is a real Z-matrix, then the following conditions are equiva-lent:(1) M is an almost non-singular M -matrix.(2) M + D is a non-singular M -matrix for any diagonal matrix D > .(3) det( M ) ≥ and det( M + D ) > det( M + D ′ ) > for any diagonal matrices such that D > D ′ > . The monotonicity of the determinant of an almost non-singular M -matrix is very important andmotivates the concept of a quasi M -matrix, which is given next.2.2. Quasi M -matrices. Here we will introduce the class of quasi M -matrices. This class ofmatrices generalizes M -matrices in a very simple way. Moreover, it has good properties that willbe very useful for our algorithm. Definition 2.10.
A real Z-matrix M is called a quasi (non-singular) M -matrix if all its properprincipal minors are nonnegative (positive). Note that M is a quasi (non-singular) M -matrix if it satisfies the condition of being an M -matrixexcept maybe for its determinant. That is, M is a quasi (non-singular) M -matrix of size n if all ofits sub-matrices of size ( n − × ( n −
1) are (non-singular) M -matrices. Example 2.11.
Let M = − −
10 1 − − − . It is not difficult to check that M is a quasi non-singular M -matrix. However, M is not an M -matrix because its determinant is equal to − . On the other hand N = M + = − −
10 1 − − − is an almost non-singular M -matrix with determinant equal to zero. Thus, by Theorem 2.9, wehave that N + D is a non-singular M -matrix for any positive diagonal matrix D . Quasi M -matrices are close to be quasi non-singular M -matrices in a similar way that M -matricesare from being non-singular M -matrices. Thus the next result can be seen as a generalization of [1,Lemma 4.1, Section 6]. Theorem 2.12.
Let M be a real Z-matrix. Then M is a quasi M -matrix if and only if M + ǫI is a quasi non-singular M -matrix for any ǫ > .Proof. ( ⇒ ) Let M i be the submatrix resulting of deleting the i th row and column. Since M is aquasi M -matrix we know that M i is an M -matrix for all i . Then, by [1, Lemma 4.1, Section 6],every M i + ǫI n − = ( M + ǫI ) i is a non-singular M -matrix for any ǫ >
0, where I n − is the identitymatrix of size n −
1. Thus, for any positive ǫ , the matrix M + ǫI is a Z-matrix with all of its propersub-matrices non-singular M -matrices. That is, M + ǫI is a quasi non-singular M -matrix for any ǫ > LGORITHMIC ASPECTS OF ARITHMETICAL STRUCTURES 7 ( ⇐ ) Conversely, if M + ǫI is a quasi non-singular M -matrix for all ǫ >
0, then ( M + ǫI ) i is anon-singular M -matrix for all i and ǫ >
0. Thus, by [1, Lemma 4.1, Section 6] M i is an M -matrixand therefore M is a quasi M -matrix. (cid:3) Before continuing we will fix some notation. If M is a matrix, let g M ( X ) be the polynomial givenby det(Diag( X ) + M ), where X is the vector of variables ( x , x , . . . , x n ). Proposition 2.13.
Let M be a square matrix of size n . Then for every s ∈ ( n ) = { , . . . , n } wehave that g M s ( X | s ) = ∂g M ∂x s , where X | s = X \ { x s } and M s is the matrix obtained by erasing the s -th row and column from M .Proof. Clearly, it is enough to prove the result for s = 1. Let M = Diag( d ) − L . Now let us notethat by the minor expansion formula of the determinant g M ( X ) = det(Diag( X + d ) − L ) = ( x + d ) g (Diag( d | ) − L ) ( X | ) + g ( x , x , . . . , x n )for some polynomial g and where d | s is the vector resulting from d by erasing the s -th entry. Since M = (Diag( d | ) − L ), then ∂g M ∂x = g M ( X | ). (cid:3) The next result is a key component of algorithms 3.3 and 3.5 given at next section and a general-ization of Theorem 2.9 to quasi non-singular M -matrices. Theorem 2.14. If M is a real Z-matrix, then M is a quasi non-singular M -matrix if and only if det( M + D ) > det( M + D ′ ) > det( M ) for every diagonal matrices such that D > D ′ > .Proof. Let
D > D ′ > E i = ( e j,k ) be the elementary matrix with e i,i = 1 and e j,k = 0 for all ( j, k ) = ( i, i ).( ⇒ ) Since M + D is a quasi non-singular M -matrix, thendet( M ǫ [ I ; I ]) = det( M [ I ; I ]) + ǫ · det( M [ I \ k ; I \ k ]) > det( M [ I ; I ])for all ǫ >
0. Thus, since D = P ni =1 d i · E i for some d i ∈ R + ,det( M + D ) > det( M ) . Moreover, using similar arguments it can be proven that det( M + D ′ + F ) > det( M + D ′ ) for anydiagonal matrix F >
0. Finally, taking F = D − D ′ > ⇐ ) Let g M ( X ) = det(Diag( X )+ M ). By hypothesis g M ( X ) is an increasing function on ( R + ∪{ } ) n and therefore every first partial derivate is positive on ( R + ∪ { } ) n . Also, g M ( X ) = X I ⊆ ( n ) det( M [ I ; I ]) · x I c , where x J = Y j ∈ J x j for all J ⊆ ( n ) . Now, we need to prove that det( M [ J ; J ]) > J ( ( n ). If | J | = n −
1, then det( M [ J ; J ]) = ∂g/∂x j (0 , . . . , > J = ( n ) \ j for some j ∈ ( n ). Now, let | J | < n − y i = x for i / ∈ J and y i = 0 for i ∈ J . If det( M [ J ; J ]) < ∂f /∂x ( y , . . . , y n ) is negative,which is a contradiction since ∂g/∂x is positive on ( R + ∪ { } ) n and therefore det( M [ J ; J ]) ≥ | J | < n −
1. Now, there exists i ∈ ( n ) such that J ( ( n ) \ i = I . Furthermore, M [ I ; I ] isan M -matrix (because is a Z-matrix and all its principal minors are nonnegative), but given thatdet( M [ I ; I ]) > M [ I ; I ] is actually a non-singular M -matrix. Therefore all the principal minorsof M [ I ; I ] are positive and in particular, det( M [ J ; J ]) > (cid:3) CARLOS E. VALENCIA AND RALIHE R. VILLAGR ´AN
Remark 2.15.
Hence, given a non-negative matrix L with zeros on the diagonal of size n and d ∈ N n + such that L d = Diag ( d ) − L is a quasi non-singular M -matrix there exists a vector d ′ ∈ N n + such that det( Diag ( d + d ′ ) − L ) ≥ . That is, ( Diag ( d + d ′ ) − L ) is an almost non-singular M -matrix.This can be summarized as that every square Z-matrix with a non-negative diagonal “aspires” tobecome an almost non-singular M -matrix. At the beginning of this section we study arithmetical structures on non-negative matrices. How-ever, can we ask what happens for arithmetical structures on an integer square matrix with zerodiagonal but possibly negative off-diagonal entries? In this scenario some things can change as wecan see in the following example.
Example 2.16.
Suppose we insist on defining arithmetical structures on integer matrices as pairsof vectors with positive integer entries satisfying condition 1.1. Let L = −
10 0 21 1 0 = | {z } L + + −
10 0 00 0 0 | {z } L − Since L + is irreducible, then it has a finite number of arithmetical structures. Moreover, it is notdifficult to check that L also has a finite number of arithmetical structures, namely; ((1 , , , (1 , , , ((1 , , , (2 , , , ((1 , , , (5 , , , ((3 , , , (1 , , , ((5 , , , (1 , , , and ((2 , , , (1 , , L , ) , its canonical arithmetical structure. Note that for each of the d-arithmetical structures of L we have that the matrix L d = Diag ( d ) − L has positive proper principalminors and determinant equal to zero. In this sense the equivalence between the properties ofthe principal minors of L d and the arithmetical structures of L established in Theorem 2.7 holds.Nevertheless, L is not a non-negative matrix and this example may be misleading as we will seenext.Now, let K a = − a = | {z } K + + − a | {z } K − , for some positive integer a . Note that K + is reducible and therefore has an infinite number of arith-metical structures. However, we have that K a has only one arithmetical structure: ((6 a, , , (1 , a, a )) .In contrast with L , when the matrix has not all the entries of K a non-positive, we do not longerhave the concept of a canonical arithmetical structure (at least not as we knew it). Moreover, notethat the proper minor ( Diag (6 a, , − K a )[ { , } ] is equal to − for every a . Hence, in this infinitefamily of examples we have lost the essence of Theorem 2.7 and remark 2.15. On the other hand, we may define the arithmetical structures of a general integer matrix with zerodiagonal L in terms of the principal minors of L d . See Section 4 for more information regarding thisdefinition. Either way, the first challenge we encounter in this scenario is to establish a finitenesscondition for the set of arithmetical structures.3. The algorithm
This section contains the main results of this article, an algorithm that computes all the arithmeticalstructures on a non-negative matrix with zero diagonal. Before presenting the algorithm let us fix
LGORITHMIC ASPECTS OF ARITHMETICAL STRUCTURES 9 some notation. Let d ∈ N n + , X = ( x , x , . . . , x n ) and f L, d ( X ) = det( Diag ( X + d ) − L ) . For simplicity we write f L ( X ) instead of f L, ( X ). Now, let coef L, d ( x a ) be the coefficient of themonomial x a = x a · · · x a n n in f L, d ( X ). The independent term of f L, d ( X ) is equal to coef L, d ( x ) = f L, d ( ), which will be denoted by c L, d . The coefficients of f L, d ( X ) that are not the independentterm are called proper coefficients . Note that the coefficients of the polynomial f L, d ( X ) are incorrespondence with the principal minors of (Diag( d ) − L ). Thus, by inspecting the polynomial f L, d ( X ) we can infer what type of (quasi) M -matrix (Diag( d ) − L ) is. In a similar manner, d isa d-arithmetical structure of L if and only if c L, d = 0 and the proper coefficients of f L, d ( X ) arepositive. Next we will show a simple example for when the condition c L, d = 0 is not enough toguarantee that a vector is a d -arithmetical structure. Example 3.1.
Let P be the path with five vertices, r = − − and L a = − − − − a − − −
10 0 0 − = Diag(1 , , a, , − A ( P ) . Since L a r = , then det( L a ) = 0 for all a ∈ N + and therefore D ( L ) ( { d | f L, d (0) = c L, d = 0 } . Recall that D ≥ ( L ) was defined as the set of d ∈ N n + such that (Diag( d ) − L ) is an almost non-singular M -matrix. Equivalently D ≥ ( L ) is the set of vectors d ∈ N n + such that all proper co-efficients of f L, d ( X ) are positive and c L, d ≥
0. Thus, by 2.14 the problem of getting an almostnon-singular M -matrix from a quasi non-singular M -matrix by adding a positive vector to the diag-onal is similar to the knapsack problem, see for instance [8] for an extensive study of the knapsackproblem.If M is a quasi non-singular M -matrix, let C ( M ) = { d ∈ N n + | ( M + Diag( d )) is an almost non-singular M -matrix } . It is not difficult to check that C ( M ) exists and is finite by Dickson’s Lemma. Now let min D ≥ ( L )be the set of all minimal elements of D ≥ ( L ), L s is the submatrix of L that results from removingthe s -th row and column. Also, for any d ∈ Z n − and 1 ≤ s ≤ n , let d s ∈ Z n be given by(3.1) d s [ i ] = d [ i ] if 1 ≤ i < s, i = s, d [ i −
1] if s < i ≤ n. Before presenting our first algorithm let us address the smaller case in the following result.
Lemma 3.2. If a, b ∈ N , then (3.2) min D ≥ (cid:18) ab (cid:19) = min (cid:26)(cid:16) d, max (cid:0) , l abd m(cid:1)(cid:17) (cid:12)(cid:12)(cid:12) d ∈ N + , d ≤ max(1 , ab ) (cid:27) , Proof.
This result is straightforward. Given a vector ( d , d ) ∈ N , the only condition needed sothat ( d , d ) ∈ min D ≥ (cid:16) ab (cid:17) , is that d d ≥ ab . (cid:3) Note that this is indeed the base case of the algorithm that follows next.
Algorithm 3.3.
Input : A non-negative square matrix L of size n with zero diagonal. Output : min D ≥ ( L ) .(1) Compute ˜ A s = min D ≥ ( L s ) for all ≤ s ≤ n .(2) Let A s = { ˜d s | ˜d ∈ ˜ A s } .(3) For δ in Q s ∈ ( n ) A s :(4) d = max (cid:8) δ [1] , . . . , δ [ n ] (cid:9) .(5) Let S = { s | coef L, d ( x s ) = 0 } and k = | S | .(6) F ind ( L, d , k ) :(7) While k > :(8) If k=1:(9) Make d [ s ] = d [ s ] + 1 and F ind ( L, d , for each s ∈ S .(10) Else :(11) Make d [ s ] = d [ s ] + 1 and F ind ( L, d , for each s ∈ S .(12) Make d [ s ] = d [ s ] + 1 and F ind ( L, d , for each s / ∈ S .(13) For d ′ ∈ min C ( Diag ( d ) − L ) :(14) “Add” d ′ + d to min D ≥ ( L ) .(15) Return min D ≥ ( L ) . At step (6) we find all minimal vectors greater than d such that all coefficients of f L, d ( X ) arepositive except, maybe, for the independent term. The function “add” at step (14) means that weadd the vector d ′ + d to the set min D ≥ ( L ) whenever it is not greater than other vector alreadyin the set. Afterwards, as erasing every vector greater than d ′ + d from the set, the minimality ofthe set is assured.Now, we are ready to prove the correctness of the Algorithm 3.3. Theorem 3.4.
Algorithm 3.3 computes the set min D ≥ ( L ) for any given non-negative matrix L with zero diagonal.Proof. We proceed by induction on the size of L . First, the case when L is a matrix of size 2 issolved by (3.2). Now, assume that the algorithm is correct for every matrix of size 1 ≤ m ≤ n − L be a non-negative matrix with zero diagonal of size n . If d ′ is a vector as given at step(4), then by 2.14 every vector d ≥ d ′ such that (Diag( d ) − L ) is an almost non-singular M -matrix isreached or found on steps (5) to (14). Therefore we only need to prove that every d ∈ min D ≥ ( L )is reachable from some vector of the form presented at step (4).Indeed, for every d ∈ D ≥ ( L ), let d | s be the vector equal to d without the s-th entry. That is, d | s [ i ] = ( d [ i ] , if 1 ≤ i ≤ s − , d [ i + 1] , if s ≤ i ≤ n − . Then for every s ∈ ( n ) d | s ∈ D ≥ ( L s ) and there exists ˜d ∈ min D ≥ ( L s ) such that ˜d ≤ d | s .Consequently, we have that max s ∈ ( n ) n ˜d s [ i ] o ≤ d [ i ] , where d s is as in (3.1), which concludes the proof. (cid:3) LGORITHMIC ASPECTS OF ARITHMETICAL STRUCTURES 11
Note that Algorithm 3.3 is not fast (not of polynomial-time), because the extended knapsackproblem of finding C ( M ) is not in general of polynomial-time. On the other hand, thanks to therecursive structure of the algorithm, we get rid of the need of checking the value of the 2 n − L . Now, we present the following algorithm that uses Algorithm 3.3. Algorithm 3.5.
Input : A non-negative square matrix L with zero diagonal. Output : D ( L ) .(1) If (L is irreducible):(2) A = min D ≥ ( L ) ,(3) D = { d ∈ A : f L, d ( ) = 0 } ,(4) Return D.(5) Elif (L has a row equal to ):(6) Return ∅ .(7) Else :(8) ’there is an infinite number of arithmetical structures’, (see Theorem 2.2).
Note that if we have a d -arithmetical structure on a matrix L , then it is very simple to get thecorresponding r -arithmetical structure. We only need to compute the kernel of (Diag( d ) − L ). Now,we are able to present the correctness of Algorithm 3.5. Corollary 3.6.
Algorithm 3.5 computes the set of arithmetical structures on any non-negativesquare matrix L with diagonal zero.Proof. It follows directly from Proposition 2.1 and Theorems 2.2, 2.7 and 3.4. (cid:3)
The next example illustrates how Algorithm 3.3 works. Moreover, it will give us a glance of itscomplexity.
Example 3.7.
Consider the graph G given in Figure 2 and let L be its adjacency matrix. v v v v
31 119 1 L = Figure 2.
The digraph G and its adjacency matrix L . Since G is a strongly connected graph, then L is an irreducible matrix. Therefore, the first step ofAlgorithm 3.5 consists of computing the sets min D ≥ ( L s ) for all the submatrices of L of size n − .It can be checked that min D ≥ ( L ) = { (1 , , , (1 , , , (2 , , , (2 , , , (1 , , , (1 , , , (3 , , , (3 , , , (3 , , } , min D ≥ ( L ) = { (1 , , , (1 , , , (1 , , } and min D ≥ ( L ) = min D ≥ ( L )= { (2 , , , (5 , , , (3 , , , (4 , , , (1 , , , (10 , , } . From this we get that the set of vectors d given at step (5) of Algorithm 3.3 is equal to (5 , , , − , (5 , , , − , (10 , , , − , (10 , , , − , (2 , , , − , (3 , , , − , (4 , , , − , (1 , , , − , (5 , , , − , (5 , , , − , (10 , , , − , (10 , , , − , (1 , , , − , (1 , , , − , (2 , , , − , (2 , , , − , (3 , , , − , (3 , , , − , (4 , , , − , (4 , , , − . where the sub-index in each vector d corresponds to the determinant of ( Diag ( d ) − L ) . Followingthe rest of algorithm 3.3 we get that min D ≥ ( A ( G )) is equal to (5 , , , (5 , , , (9 , , , (9 , , , (6 , , , (6 , , , (6 , , , (5 , , , (5 , , , (5 , , , (5 , , , (5 , , , (5 , , , (7 , , , (7 , , , (5 , , , (2 , , , (2 , , , (2 , , , (3 , , , (2 , , , (2 , , , (2 , , , (3 , , , (3 , , , (9 , , , (1 , , , (1 , , , (1 , , , (18 , , , (10 , , , (10 , , , (14 , , , (14 , , , (12 , , , (12 , , , (12 , , , (11 , , , (11 , , , (11 , , , (11 , , , (10 , , , (10 , , , (10 , , , (10 , , , (10 , , , (23 , , , (23 , , , (16 , , , (16 , , , (14 , , , (14 , , , (12 , , , (12 , , , (11 , , , (11 , , , (10 , , , (10 , , , (4 , , , (4 , , , (4 , , , (18 , , , (18 , , , (5 , , , (5 , , , (6 , , , (6 , , , (6 , , , (6 , , , (7 , , , (7 , , , (8 , , , (8 , , , (5 , , , (5 , , , (9 , , , (9 , , , (4 , , , (4 , , , (10 , , , (10 , , , (10 , , , (10 , , , (11 , , , (11 , , , (12 , , , (12 , , , (13 , , , (13 , , , (14 , , , (14 , , , (15 , , , (15 , , , (16 , , , (16 , , , (18 , , , (18 , , , (23 , , , (23 , , , (36 , , , (36 , , , (1 , , , (1 , , , (1 , , , (1 , , , (1 , , , (1 , , , (5 , , , (5 , , , (3 , , , (3 , , , (3 , , , (3 , , , (2 , , , (2 , , , (2 , , , (2 , , , (2 , , , (2 , , , (9 , , , (9 , , , (3 , , , (3 , , , (4 , , , (4 , , , Thus, min D ≥ ( L ) has elements and of them are d-arithmetical structures on L . The set D ( L ) is listed below. (9 , , ,
3) (9 , , ,
2) (6 , , ,
3) (5 , , ,
3) (5 , , ,
6) (5 , , , , , ,
2) (3 , , ,
2) (2 , , ,
3) (3 , , ,
3) (3 , , ,
2) (9 , , , , , ,
2) (1 , , ,
3) (1 , , ,
2) (18 , , ,
3) (10 , , ,
3) (10 , , , , , ,
3) (12 , , ,
5) (18 , , ,
5) (18 , , ,
1) (5 , , ,
18) (5 , , , , , ,
5) (6 , , ,
1) (6 , , ,
9) (6 , , ,
1) (9 , , ,
6) (9 , , , , , ,
33) (10 , , ,
1) (12 , , ,
15) (12 , , ,
1) (18 , , ,
9) (18 , , , , , ,
7) (36 , , ,
1) (1 , , ,
4) (1 , , ,
1) (1 , , ,
6) (1 , , , , , ,
4) (3 , , ,
1) (2 , , ,
5) (2 , , ,
1) (2 , , ,
9) (2 , , , , , ,
4) (9 , , ,
1) (3 , , ,
6) (3 , , ,
1) (4 , , ,
7) (4 , , , L with diagonal zero we have an algorithm that computesall the integer solutions of the polynomial equation det( Diag ( X ) − L ) = 0. Thus, it is natural toask: When a polynomial is the determinant of a matrix with diagonal X ? In the following we willsee that not every polynomial is the determinant of a matrix of the form Diag ( X ) − L. Moreover, we show that the set of monic (every term) square-free polynomials that are the deter-minant of a matrix
Diag ( X ) − L is in some sense very small. Firstly, it is clear that if a polynomialis equal to det( Diag ( X ) − L ), then it must be monic (the coeffcient of the monomial x x . . . x n isalways 1) and (every term) square-free. Therefore we restrict to monic polynomials with every termsquare-free. For the rest of this section we will simply call them monic and square-free polynomials. LGORITHMIC ASPECTS OF ARITHMETICAL STRUCTURES 13
Let Z [ X ] ∗ = (cid:8) f ∈ Z [ X ] (cid:12)(cid:12) f is monic and square-free (cid:9) / ( ∼ ) , where f ∼ g if there exists d ∈ Z n such that f ( X + d ) = g ( X ). We consider two polynomialsequivalent because when one can be obtained as an evaluation of the other is because its integersolutions are essentially the same. Note that Z [ X ] ∗ ≃ (cid:8) f ∈ Z [ X ] (cid:12)(cid:12) f is monic square-free with coef (cid:18) Q ni =1 x i x j (cid:19) = 0 for all 1 ≤ j ≤ n (cid:9) . Also, let MP [ X ] = (cid:8) f ∈ Z [ X ] ∗ (cid:12)(cid:12) f = det(Diag( X ) − L ) for some matrix L with zero diagonal (cid:9) .Clearly MP [ X ] ⊆ Z [ X ] ∗ . If | X | = 2, we can easily check equality holds, that is, MP [ x , x ] = Z [ x , x ] ∗ . In a similar way,for | X | = 3 we will prove the following result. Proposition 3.8. If X = ( x , x , x ) , then MP [ X ] = (cid:8) f = x x x + a x + a x + a x + b ∈ Z [ X ] ∗ (cid:12)(cid:12) b = a a a − n n ∈ Z (cid:9) . Proof.
Let A = a , a . a . a , a , a , be an integer matrix with zero diagonal. Thendet( Diag ( x , x , x ) − A ) = x x x − a , a , x − a , a , x − a , a , x +( − a , a , a , − a , a , a , )Now, let us set a = − a , a , , a = − a , a , , a = − a , a , and b = ( − a , a , a , − a , a , a , ).Therefore, b = a a a a , a , a , − a , a , a , . We know that both b and a , a , a , are integer numbers.Then a , a , a , is a divisor of a a a , since a a a a , a , a , is also an integer. Now, we can concludethat b = a a a n − n , where n divides a a a . (cid:3) Note that b ∈ Z if and only if n divides a a a . If a a a = 0, then there is a finite set of b ′ s on Z such that f = x x x + a x + a x + a x + b is in MP [ x , x , x ] and therefore MP [ X ] ( Z [ X ] ∗ forall n ≥
3. On the other hand, if a a a = 0, then x x x + a x + a x + a x + b is in MP [ x , x , x ]for every b ∈ Z . Furthermore, if f comes from a matrix, then f = det x − n a n a n x − n − n a n x , where n = n n n and n i | a i (here we are considering every integer as a “divisor” of 0).Next example will be helpfull to illustrate this. Example 3.9. If g = x x x − x + 2 x + 3 x + b , then b = − n − n where n ∈ Div(114) = ±{ , , , , , , , } . Which implies that b ∈ ±{ , , , } . It is not difficult to check by proposition 3.8 that f ( x , x , x ) = x x x − x + 2 x + 3 x − / ∈ MP [ x , x , x ] . When n ≥ f . Moreover, thegap between MP [ X ] and Z [ X ] ∗ grows as n grows.In order to finish this section we present some computational evidence for conjecture 3.10. Conjecture 3.10. [3, Conjecture 6.10] If G is a simple graph with n vertices, then (cid:12)(cid:12)(cid:12) A ( P n ) (cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12) A ( G ) (cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12) A ( K n ) (cid:12)(cid:12)(cid:12) , where P n and K n are the path with and the complete graph with n vertices respectively. Graph G |A ( G ) | Graph G |A ( G ) | Graph G |A ( G ) | (cid:12)(cid:12)(cid:12) A ( K ) (cid:12)(cid:12)(cid:12) = 2025462, (cid:12)(cid:12)(cid:12) A ( K ) (cid:12)(cid:12)(cid:12) = 1351857641 and (cid:12)(cid:12)(cid:12) A ( K ) (cid:12)(cid:12)(cid:12) ≃ . × . Moreover thelargest d i such that d ∈ D ( K n ) is given by the sequence a ( n ) = a ( n − a ( n −
1) + 1) −
1. Forinstance the highest d i in a d-arithmetical structure of K is about 1 . × . A deeper studyof the possible value of the largest entry of an arithmetical structure on the complete graph isconducted in [7]. LGORITHMIC ASPECTS OF ARITHMETICAL STRUCTURES 15 Arithmetical structures on dominated polynomials.
Inspired by Algorithms 3.3 and 3.5, at this section we generalize some of the ideas presented before(for the polynomials which are the determinant of a matrix with variables in the diagonal) to thenew class of integer polynomials, which we called dominated. Some concepts are preserved in thisnew setting and others are not. For instance, the concept of d -arithmetical structure is generalizedeasily. However, we are not allow to find a good definition for an r -arithmetical structure.An algorithm similar to Algorithm 3.5 that computes the d -arithmetical structures for dominatedpolynomials is presented. Unfortunately, this algorithm does not give an efficient way to computeall integer solutions of a dominated polynomial.First, since the set of solutions of the product of two polynomials can be easily obtained in functionof the solutions of each polynomial in this setting, then we can assume that the polynomial isirreducible. Moreover, using some simple changes of variable (of the type x i = − x i ) we can getall the integer solutions from the positive ones and therefore we can restrict to positive integersolutions only. From now on all the polynomials that we consider are dominated and square-free.We continue by introducing dominated polynomials.4.1. Dominated polynomials.
Given a set of monomials M in Z [ x , x , . . . , x n ], a monomial p ∈ M is called a dominant monomial of M whenever is divided by every monomial in M (anyother monomial in M is a factor of p ). Is not difficult to check that if M has a dominant monomial,then it is unique. Let M f be the set of monomials with non-zero coefficient on the polynomial f . Definition 4.1.
A polynomial f ∈ Z [ x , . . . , x n ] is dominated when M f has a dominant monomial. Let f be a dominated polynomial and p f ∈ M f be its dominant monomial. If p f is square-free,then f is called a square-free dominated polynomial. Moreover, if f is a polynomial such that everyvariable appears at least once, then f is a dominated square-free polynomial if and only if n Y i =1 x i ∈ M f , which is precisely its dominant monomial. Let f d ( X ) denote f ( X + d ).Now we can proceed to give a definition of a d -arithmetical structure of a polynomial. Definition 4.2.
Let f ∈ Z [ X ] be an irreducible square-free dominated polynomial with its leadingcoefficient positive. An arithmetical structure of f is a vector d ∈ N n + such that f ( d ) = 0 and allthe non-constant coefficients of f d ( X ) are positive. If f does not have its leading coefficient positive, then it does not have any arithmetical structures.On the other hand, since either f or − f has its leading coefficient positive, then we can assumethat f has positive leading coefficient. From now on, let us assume that the leading coefficient isalways positive unless the contrary is stated.When the polynomial f ∈ Z [ X ] is not irreducible, that is f = Q si =1 f i for some irreducible square-free polynomials f i , we need to introduce some extra notation. Since f is square-free, the set ofvariables of the f i ’s does not intersect and therefore each f i is a dominated polynomial. Thus,given d ∈ Z n , let d ( f i ) be the vector with the entries of d that corresponds to the variables of f i .In the general case of reducible square-free polynomials an arithmetical structure is a d ∈ Z n suchthat d ( f i ) is an arithmetical structure of at least one of the f i and the non-constant coefficientsof f i, d ( f i ) ( X ) are positive and the constant coefficient is non-negative for all i . Thus when f isreducible square-free polynomial, then it has an infinite number of arithmetical structures. Definition 4.3.
Given a square-free dominated polynomial f on n variables, let D ( f ) = { d ∈ N n + | d is an arithmetical structure of f } . It is not difficult to check that this definition generalizes the one given in Section 2. More precisely, D ( L ) = D ( f L ) for any non-negative matrix with zero diagonal L .Defining an r -arithmetical structure of an integer square-free dominated polynomial is a moredifficult task. For one hand, the r -arithmetical structures of L and L t are equal if and only if L issymmetric. And for the other hand, f L ( X ) = f L t ( X ) for any L ∈ M n ( Z ) because the determinantof a matrix is invariant under the transpose, that is, det( L ) = det( L t ). Moreover, if M is a matrixwithout rows or columns equal to zero, then D ( L ) = D ( L t ). That is, the polynomial f L ( X ) doesnot distinguish between L and L t . However r -arithmetical structures of L and L t are not equalwhen L is not symmetric. Therefore in general we may not try to extract the information of the r -arithmetical structures from f L ( X ). Example 4.4. If M = (cid:18) (cid:19) , then f L ( x , x ) = f L t ( x , x ) = x x − and therefore A ( L ) = { ((1 , , (1 , , ((3 , , (1 , } and A ( L t ) = { ((1 , , (3 , , ((3 , , (1 , } . Thus D ( f L ) = { (1 , , (3 , } = D ( f L t ) and R ( f L ) = { (1 , , (1 , } 6 = { (1 , , (3 , } = R ( f L t ) . Remark 4.5.
If a polynomial f in MP [ X ] is irreducible, then it comes from an irreducible matrix. Since a symmetric Z -matrix M is an almost non-singular M -matrix with det( M ) = 0 if and onlyif there exists r > Adj ( M ) = | K ( M ) | r t r > , where ker Q ( M ) = h r i and K ( M ) is the critical group of M , see [3, Proposition 3.4]. Then is factibleto define the critical group of a d -arithmetical structure of a polynomial f as | K ( f, d ) | = gcd(coef f d ( X ) ( x ) , . . . , coef f d ( X ) ( x n )) . Given any non-negative matrix with zero diagonal L such that every of its rows are different from , then ( L , ) is the canonical arithmetical structure of L . In general for polynomials in Z [ X ] ∗ wecan not recover the concept of canonical arithmetical structure. Furthermore, some polynomialsare extremal in the sense that they have very few arithmetical structures. We illustrate this ideaat the next example. Example 4.6.
Returning to the polynomial of example 3.9, f ( x , x , x ) = x x x − x + 2 x + 3 x − / ∈ MP [ x , x , x ] . Evaluating, it is easy to see that ( d , d , d ) N is an arithmetical structure of f if and only if d d − ≥ and ( d d − d + 2 d + 3 d = 23 . Thus we have that D ( f ) = { (1 , , } . A follow up problem would be to study this type of polynomials,where we have a single d-arithmetical structure. An algorithm for the polynomial case.
At this subsection we extend Algorithms 3.3and 3.5 to find arithmetical structures of a square-free dominated polynomial with integer coeffi-cients. First, let D ≥ ( f ) = n d ∈ N n + (cid:12)(cid:12)(cid:12) all non-constant coefficients of f d ( X ) positive and f ( d ) ≥ o . Now, we are ready to present our algorithm.
LGORITHMIC ASPECTS OF ARITHMETICAL STRUCTURES 17
Algorithm 4.7.
Input : A square-free dominated polynomial f over Z . Output : min D ≥ ( f ) and D ( f ) .(1) If f is irreducible:(2) Let ∂ s f = ∂f∂x s for all ≤ s ≤ n .(3) Compute A s = min D ≥ ( ∂ s f ) for all ≤ s ≤ n .(4) Let ˜ A s = { ˜d s | d s ∈ A s } .(5) For δ in Q ns =1 ˜ A s :(6) d = max { δ [1] , . . . , δ ( n ) } .(7) Let S = { s | coef d ( x s ) = 0 } and k = | S | .(8) F ind ( f, d , k ) :(9) While k > :(10) If k = 1 :(11) Make d [ s ] = d [ s ] + 1 and F ind ( f, d , for each s ∈ S .(12) Else :(13) Make d [ s ] = d [ s ] + 1 and F ind ( f, d , for each s ∈ S .(14) Make d [ s ] = d [ s ] + 1 and F ind ( f, d , for each s / ∈ S .(15) For d ′ ∈ min C ( f ( X + d )) :(16) Add d ′ + d to min D ≥ ( f ) .(17) Return min D ≥ ( f ) and D ( f ) = { d ∈ min D ≥ ( f ) | f ( d ) = 0 } .(18) Else: ( f is reducible)(19) Compute A s = min D ≥ ( f s ) for all irreducible factors f s of f .(20) “Choose all possible combinations”. Remark 4.8.
Since f is square-free, its irreducible factors do not share variables. Therefore, step (20) refers to choose an element in D ( f i ) for some irreducible factor f i of f . Choose an elementin D ≥ ( f j ) for some j = i and merging these vectors to get an arithmetical structure of f . We are ready to prove the correctness of Algorithm 4.7. The prove will be similar to the one givenfor Algorithm 3.3. Thus we begin by extendying Lemma 3.2 for the polynomial case.
Lemma 4.9. If a, b , b , c ∈ Z , a ≥ and f = ax x + b x + b x + c , then min D ≥ ( f ) = min (cid:26)(cid:18) d, max (cid:16) d +2 , l − ( c + b d ) ad + b m(cid:17)(cid:19) (cid:12)(cid:12)(cid:12) d ∈ N + , d +1 ≤ d ≤ max (cid:16) d +1 , l − ( c + b d +2 ) ad +2 + b m(cid:17)(cid:27) , where d +1 = max(1 , ⌈ − b a ⌉ ) and d +2 = max(1 , ⌈ − b a ⌉ ) .Proof. A vector d = ( d , d ) ∈ Z is in D ≥ ( f ) if and only if d ≥ , ad + b ≥ d ≥ , ad + b ≥ ad d + b d + b d + c ≥ . We set d +1 = max(1 , ⌈ − b a ⌉ ) and d +2 = max(1 , ⌈ − b a ⌉ ). It is clear that if d ∈ D ≥ ( f ), then d ≥ ( d +1 , d +2 ). On the other hand, if ( d , d ) ≥ ( d +1 , d +2 ), then the only condition left for d to be in D ≥ ( f ) is 4.1. Therefore, if ad +1 d +2 + b d +1 + b d +2 + c ≥ , then min D ≥ ( f ) = { ( d +1 , d +2 ) } . Henceforth, let us assume that(4.2) ad +1 d +2 + b d +1 + b d +2 + c < ≤ − ad d +2 + b d + b d +2 + c < . Thus d +1 ≤ d < − ( c + b d +2 ) ad +2 + b and in order to fulfill condition 4.1, we have that d ≥ − ( c + b d ) ad + b . Also note that max( d +2 , − ( c + b d ) ad + b ) = − ( c + b d ) ad + b by 4.3. Then min n ( d , ⌈ − ( c + b d ) ad + b ⌉ ) | d +1 ≤ d ≤ ⌊ − ( c + b d +2 ) ad +2 + b ⌋ o ⊆ min D ≥ ( f ). Finally, if(4.4) ad d +2 + b d + b d +2 + c ≥ , then we have that max( d +2 , − ( c + b d ) ad + b ) = d +2 and d ≥ − ( c + b d +2 ) ad +2 + b . Thus { ( ⌈ − ( c + b d +2 ) ad +2 + b ⌉ , d +2 ) } = min { d ∈ D ≥ ( f ) | } . We conclude thatmin D ≥ ( f ) = min n { ( d, ⌈ − ( c + b d ) ad + b ⌉ ) | d +1 ≤ d ≤ ⌊ − ( c + b d +2 ) ad +2 + b ⌋} ∪ { ( ⌈ − ( c + b d +2 ) ad +2 + b ⌉ , d +2 ) } o if 4.2 holds { ( d +1 , d +2 ) } otherwise . Clearly, this can be restated so that we have the result. (cid:3)
Remark 4.10.
Note that D ≥ ( f ) is an infinite set. Moreover, we have monotonicity of f as into Theorem 2.14. More precisely, if g ( x , x ) = f ( x + d +1 , x + d +2 ) has positive non-constantcoefficients, then g ( x + ǫ ′ , x + ǫ ′ ) > g ( x + ǫ , x + ǫ ) > g ( x , x ) for every ( ǫ ′ , ǫ ′ ) > ( ǫ , ǫ ) > . Example 4.11.
Let f = f ( x , x ) = 2 x x − x − x +16 and let d +1 and d +2 be as in Lemma 4.9.It is not difficult to check that ( d +1 , d +2 ) = (6 , and min D ≥ ( f ) = min (cid:26)(cid:16) d, max (cid:0) , l − (16 − d )2 d − m(cid:1)(cid:17)(cid:12)(cid:12)(cid:12) d ∈ N + , ≤ d ≤ (cid:27) = min (cid:26) (6,13),(7,9),(8,7),(9,6),(10,6),(11,6),(12,5),(13,5),(14,5),(15,5),(16,5),(17,5),(18,5),(19,5),(20,5),(21,5),(22,5),(23,5),(24,4) (cid:27) = (cid:26) (6,13),(7,9),(8,7),(9,6),(12,5),(24,4) (cid:27) . And therefore D ( f ) = { (6 , , (24 , } . Now we proceed to prove that the Algorithm 4.7 is correct.
Theorem 4.12.
Algorithm 4.7 computes the sets min D ≥ ( f ) and D ( f ) for any square-free domi-nated polynomial f ∈ Z [ X ] ∗ .Proof. First, note that induction on the size of L and the n − X + d ) − L )correspond to induction on the number of variables in X and the first partial derivatives of f respectively. Thus, if f ( X ) = f L, d ( X ) for some non-negative matrix L , then by Proposition 2.13the result follows by similar arguments of those given in Theorem 3.4. LGORITHMIC ASPECTS OF ARITHMETICAL STRUCTURES 19
In general, we can also proceed by induction, but now on the number of variables in X . When | X | = 2 and f = f ( X ) is a square-free dominated polynomial with positive leading coefficient.Assume without lost of generality that X = { x , x } . Then f = ax x + b x + b x + c and byLemma 4.9, min D ≥ ( f ) is determined.Now assume that the result holds for every polynomial on 2 ≤ k ≤ n − f ( x , . . . , x n )be an integer square-free dominated polynomial with positive leading coefficient. Given any vector d ′ in step (6) every coefficient of every monomial of degree at least 2 in f ( x + d ′ , . . . , x n + d ′ n )is positive. Also every coefficient of every monomial of degree one in f ( x + d ′ , . . . , x n + d ′ n ) isnon-negative. Whereas the independent term can be negative. But in steps (7) through (15) it isshown how to find the first vector d ≥ d ′ such that the independent term turns non-negative andthe coefficients of the degree one monomials are positive too. That is, d ∈ D ≥ ( f ).At step (16) we proceed in the same manner as in Algorithm 3.3 to produce the set of minimalelements of D ≥ ( f ). Thus, we need to prove that every d ∈ min D ≥ ( f ) is reachable from somevector of the form given in step (6). Again, in a similar way as in the proof of theorem 3.4. Now,for every d ∈ D ≥ ( f ), let d | s be the vector equal to d without the s-th entry. That is, d | s [ i ] = ( d [ i ] , if 1 ≤ i ≤ s − , d [ i + 1] , if s ≤ i ≤ n − . Then for every s ∈ ( n ) we have that d | s ∈ D ≥ (cid:18) ∂f∂x s (cid:19) and there exists ˜d ∈ min D ≥ (cid:18) ∂f∂x s (cid:19) suchthat ˜d ≤ d | s . Thus, max s ∈ ( n ) n ˜d s [ i ] o ≤ d [ i ] , where d s is as in (3.1). Therefore the algorithm computes min D ≥ ( f ) and then is clear that itcomputes D ( f ) by definition. (cid:3) We illustrate the geometry of Lemma 4.9 through Example 4.11, see Figure 4.2. Let us denote thegreen region as P G . Note that P G corresponds to D ≥ ( f ) since it is the portion of the N + -grid“above” ( d +1 , d +2 ) and such that f ≥
0. More precisely, D ≥ ( f ) = P G ∩ N . Furthermore, it isnot difficult to see that if g is a polynomial of degree n then D ≥ ( g ) = P ∩ N n + , where P is anunbounded n-dimensional polytope.The next example illustrates how Algorithm 4.7 works on a polynomial not in MP [ X ]. Example 4.13.
Let f = x x x − x + 2 x + 3 x − be the irreducible polynomial given in Example 4.6. Step (2) of Algorithm 4.7 gives us ∂ f = x x − ∂ f = x x + 2 ∂ f = x x + 3 . From step (3) and Lemma 4.9 we get that min D ≥ ( ∂ f ) = min D ≥ ( ∂ f ) = { (1 , } and min D ≥ ( ∂ f ) = { (1 , , (19 , , (2 , , (10 , , (3 , , (7 , , (4 , , (5 , } . Continuing with steps (4) to (6) we have the following set of d ′ s to search, Π = (cid:26) (1 , ,
19) (1 , ,
10) (1 , ,
7) (1 , , , ,
1) (1 , ,
2) (1 , ,
3) (1 , , (cid:27) . Note that f d ( X ) has positive independent term for almost every vector d ∈ Π , except for (1 , , .That is, only the vector (1 , , has the chance to be an arithmetical structure of f . Indeed, since f (1 , , ( X ) = x x x + 4 x x + 5 x x + x x + x + 6 x + 8 x + 0 , then D ( f ) = { (1 , , } . x x d +2 ←− c + b d +1 ad +1 + b d +1 ↓− c + b d +2 ad +2 + b Figure 3.
The blue line represents the curve f = 2 x x − x − x + 16 = 0 for x ≥ . D ≥ ( f ).We recall that if f is a square-free dominated polynomial without any arithmetical structure, thenthis does not implies that f = 0 has not integer solutions. Example 4.14.
Let g = x x + 17 x − x + 27 . By Lemma 4.9 we have that min D ≥ ( g ) = { (13 , } . On the other hand, since g (13 ,
1) = 249 , then D ( g ) = ∅ . Nevertheless g = 0 has sixteen differentsolutions in Z . Moreover four of them are solutions in N , namely { (1 , , (5 , , (9 , , (11 , } . None of them found by the algorithm. Because the condition of having all non-constant coefficientspositive is not fulfilled by any of them. For instance note that f ( x +11 , x +214) = x x +231 x − x . Integer solutions of dominated polynomials
We will finish by exploring some ideas to obtain all the integer solutions of a dominated polynomialin an efficient way. This suggests that what we have developed so far is useful beyond findingarithmetical structures. However, none of these methods are useful in general.Even though the set min D ≥ ( g ) does not necessarily contain all its positive integer solutions ofthe polynomial g . As the next example illustrates, in some cases the coefficients of g d ( X ) andmin D ≥ ( g ) gives us enough information to to get all the integer solutions of a polynomial. Example 5.1.
Following last Example 4.14 we have that for g ( x , x ) = x x + 17 x − x + 27 , min D ≥ ( g ) = { (13 , } . Thus, g ( d ) = 0 for all d ≥ (13 , . Moreover, it is not difficult to check that g ( x + 13 , x + 1)) = x x + 18 x + x + 236 and therefore its coefficients are positive. Therefore it only remains to checkwhen vectors in { ( d , d ) ∈ N (cid:12)(cid:12)(cid:12) d ≤ and d ≥ } are or not solutions of g .First, since the coefficient of x in g ( x + 12 , x + 1) = x x + 18 x + 231 = 0 is equal to zero, then g ( d ) = 0 for all d with d = 12 and d ≥ . In a similar way, since g ( x + 11 , x + 1) = x x + 18 x − x + 213 LGORITHMIC ASPECTS OF ARITHMETICAL STRUCTURES 21 it is not difficult to see that the line segment { (11 , a ) | a ∈ N + } contains at most one solution of g =0 , namely (11 , . Following this procedure we have that g ( x +10 , x +1) = x x +18 x − x +195 ,which is the line segment { (10 , a ) | a ∈ N + } and does not contain any solution of g = 0 . Finally,in the other cases we get that (9 , , (5 , and (1 , are the other positive integer solutions of g .That is, (11 , , , (5 , and (1 , are the only positive integer solutions of g . Remark 5.2.
In the general case when for | X | = k + 1 ≥ instead of looking the solution in a linesegment we need to search in subsets of k -dimensional hyperplanes. Which can be done by usingthe same techniques presented here, but in a problem with one dimension less. Unfortunately, this technique does not work in general. We know that arithmetical structures ofirreducible dominated square-free polynomials generalizes the concept for irreducible non-negativematrices with zero diagonal by Theorem 2.7. On the other hand, if we consider irreducible matriceswith some negative off-diagonal entries and diagonal zero, then having a positive vector in the kerneldoes not implies positive principal minors.
Example 5.3.
Resuming Example 2.16, consider the matrix L = −
10 0 21 1 0 and K a = − a . Taking d = (6 a, , , r = (1 , a, a ) , we can see that r > and det( Diag ( d ) − K a ) = 0 .However, as we checked in Example 2.16, not all its principal minors are positive. Furthermore,taking f K a := f K a ( x , x , x ) = det( Diag ( x , x , x ) − K a ) ,f K a = x x x − x + 6 a for every a ∈ N + . Therefore D ≥ ( f K a ) = { (1 , , , (1 , , , (1 , , } and D ( f K a ) = ∅ for every positive integer a . However, applying the previous heuristic given inExample 5.1, we can only find the positive solutions (6 a, , for each a ∈ N + given in Example 2.16. The next example shows that the procedure used before not always get all the positive integerssolutions even of a dominated polynomial. Let B an irreducible integer square matrix with somenegative off-diagonal entries and diagonal zero. Defining the arithmetical structures on B in termsof the non-negativity of its minors seems to be the best option. In this way, we would be gen-eralizing the concepts of almost non-singular (quasi non-singular) M -matrices to almost (quasi)P-matrices. The latter refers to the set of real matrices with positive proper principal minors andzero (indistinct) determinant. Example 5.4.
Applying Algorithm 4.7 to f ( x, y, z ) = xyz − x + 8 y − z − we get that D ( f ) = { (3 , , , (5 , , , (7 , , , (13 , , , (15 , , , (75 , , , (119 , , , (235 , , } . Note that { (1 , , , (1 , , , (1 , , , (13 , , , (89 , , } are solutions of f which are not d -arithmetical structures. However, starting from the arithmetical structure (3 , , and searching inset { (1 , a, b ) | a ≥ , b ≥ } as in previous Example 2.16 we get the solutions (1 , , ≤ (1 , , ≤ (1 , , . In order to finish, we present an example where even we can not find all the positive integersolutions.
Example 5.5.
Consider f = zx x y y − x x y y + x x − y y − . Thus the positive solutionsof f ( z = 1) = x x − y y − contains the positive solutions of the Pell’s equation (5.1) x − y = 1 . It is well known that the solutions of a Pell’s equation are an infinite strictly increasing sequenceof vectors. Moreover, if { ( α n , β n ) | n ∈ N } are the solution of (5.1) ordered in increasing order, thenthey satisfy the following recurrent relation α k +1 = 16 α k − α k − and β k +1 = 16 β k − α k − for k ≥ . More precisely, in our case we get the solutions (8 , , (127 , , (2024 , , (32257 , , (514088 , , (8193151 , , (130576328 , and so on . . . Thus, even when our methods are sufficient to find the arithmetical structures of a square-freedominated polynomial, in general it can not find all the positive integer solutions of a monic squarefree polynomial, for instance the solutions of a Pell’s equation. Therefore it is clear that our methodscan not give us all integer solutions in general.
Defining arithmetical structures for dominated polynomials, not necessarily square-free, is verysimple. More precisely, let F ( x , . . . , x m ) = 0 be a dominated Diophantine equation and let δ ( i )be the maximum exponent of x i in any term of F . Then let f ( x , . . . , x δ (1) , . . . , x m , . . . , x m δ ( m ) ) = 0 , be a dominated square-free Diophantine equation such that f ( x , . . . , x , . . . , x m , . . . , x m ) = F ( x , . . . , x m ) . Thus D ( F ) := n ( d , . . . , d m ) | d ′ ∈ D ( f ) such that d ′ i = · · · = d ′ i δ ( i ) for every i = 1 , . . . , m o . Similarly for D ≥ ( F ). For instance, let a ∈ N + and F ( x, y ) = xy − x + 6 a , then min D ≥ ( F ) = { (1 , } and D ( F ) = ∅ by the second part of Example 5.1. Furthermore, we can find the onlysolution for F=0 in N + × N + , x = 6 a and y = 1. Finally, considering D ≥ α ( F ) = (cid:8) d ∈ N (cid:12)(cid:12) F ( X + d ) has positive non-constant coefficients and F ( d ) ≥ α (cid:9) we can address solutions of the equation F = α for any α ∈ R . Therefore finding the arithmeticalstructures (and general solutions for that matter) of dominated Diophantine polynomials is a specialcase of square-free dominated Diophantine polynomials. References [1] Berman, A. and Plemmons, R. J., Nonnegative matrices in the mathematical sciences, vol. 9, 1994, Siam.[2] B. Braun, H. Corrales, S. Corry, L. D. Garc´ıa Puente, D. Glass, N. Kaplan, J. L. Martin, G. Musiker, and C. E.Valencia, Counting arithmetical structures on paths and cycles, Discrete Math. 341 (2018), no. 10, 2949–2963.MR 3843283[3] Corrales, H. and Valencia, C. E., Arithmetical structures on graphs, Linear Algebra and its Applications 536(2018) 120-151.[4] Corrales, H. and Valencia, C. E., Arithmetical structures on graphs with connectivity one, J. Algebra Appl. 17(8) (2018) 1850147.[5] Eisentrager, K., Miller, R., Park, J. and Shlapentokh, A., As easy as Q : Hilbert’s tenth problem for subrings ofthe rationals and number fields. Transactions of the American Mathematical Society, 369(11) (2017), 8291-8315[6] Godsil, C. and Royle, G., Algebraic Graph Theory, Graduate Texts in Mathematics, vol. 207, Springer-Verlag,New York, 2001.[7] Harris, Z. and Louwsma J., On Arithmetical Structures on Complete Graphs, arXiv:1909.02022, preprint.[8] Kellerer, H., Pferschy, U. and Pisinger, D., Knapsack problems, Springer-Verlag, Berlin, 2004.[9] Dickson, L. E., Finiteness of the odd perfect and primitive abundant numbers with n distinct prime factors,Amer. J. Math. 35 (4) (1913) 413–422.[10] Lorenzini, D.J., Arithmetical graphs. Math. Ann. 285 (1989), no. 3, 481–501.[11] Matiyasevich, Yu. V., The Diophantineness of enumerable sets. Dokl. Akad. Nauk SSSR, 191:279– 282, 1970. LGORITHMIC ASPECTS OF ARITHMETICAL STRUCTURES 23 [12] Poonen, B., Hilbert’s Tenth Problem and Mazur’s Conjecture for Large Subrings of Q . Journal of the AmericanMathematical Society, 16(4) (2003), 981-990.[13] Robinson, J., Definability and decision problems in arithmetic. Journal of Symbolic Logic, 14(2) (1949), 98-114.[14] OEIS Foundation Inc. (2019), The On-Line Encyclopedia of Integer Sequences, http://oeis.org/A002967[15] Smart, N., The Algorithmic Resolution of Diophantine Equations: A Computational Cookbook. London Math-ematical Society Student Texts, 41. Cambridge University Press, Cambridge, 1998.[16] Valencia, C. E. and Villagr´an, R. R., Arithmetical structures on graphs with twins, manuscript. Email address , C. E. Valencia: [email protected], [email protected]
Email address , R. R. Villagr´an: [email protected], [email protected]@math.cinvestav.mx, [email protected]