Abstract. Let A be a connected integer symmetric matrix, i.e., A = (aij) ∈ Mn(Z) for some n, A = AT, and the underlying graph (vertices corresponding to rows, with vertex i joined to vertex j if aij 6 = 0) is connected. We show that if all the eigenvalues of A are strictly positive, then tr(A)  ≥ 2n − 1. There are two striking corollaries. First, the analogue of the Schur-Siegel-Smyth trace problem is solved for characteristic polynomials of connected inte-ger symmetric matrices. Second, we find new examples of totally real, separa-ble, irreducible, monic integer polynomials that are not minimal polynomials of integer symmetric matrices

Aguirre

Borwein

Dobrowolski

Estes

Flammang

James McKee

Liang

McKee

Pavlo Yatsyna

Schur

Siegel

Smyth

Crossref

Linear Algebra and its Applications

A trace bound for positive definite connected integer symmetric matrices

Let A be a connected integer symmetric matrix, i.e., A=(a_ij)∈Mn(Z) for some n, A=A^T, and the underlying graph (vertices corresponding to rows, with vertex i joined to vertex j if a_ij≠0) is connected. We show that if all the eigenvalues of A are strictly positive, then tr(A)&gt;=2n−1.  There are two striking corollaries. First, the analogue of the Schur–Siegel–Smyth trace problem is solved for characteristic polynomials of connected integer symmetric matrices. Second, we find new examples of totally real, separable, irreducible, monic integer polynomials that are not minimal polynomials of integer symmetric matrices.  <br/

McKee, James

Yatsyna, Pavlo

Royal Holloway - Pure

A TRACE BOUND FOR POSITIVE DEFINITE CONNECTEDINTEGER SYMMETRIC MATRICESJAMES MCKEE AND PAVLO YATSYNAAbstract. Let A be a connected integer symmetric matrix, i.e., A = (aij) ∈Mn(Z) for some n, A = AT , and the underlying graph (vertices correspondingto rows, with vertex i joined to vertex j if aij 6= 0) is connected. We showthat if all the eigenvalues of A are strictly positive, then tr(A) ≥ 2n− 1.There are two striking corollaries. First, the analogue of the Schur-Siegel-Smyth trace problem is solved for characteristic polynomials of connected inte-ger symmetric matrices. Second, we find new examples of totally real, separa-ble, irreducible, monic integer polynomials that are not minimal polynomialsof integer symmetric matrices.1. Introduction and statement of resultsLet A = (aij) be an n × n symmetric matrix, with entries in Z (an integersymmetric matrix ). It will be convenient to associate to such a matrix its underlyinggraph, whose vertices 1, . . . , n correspond to the rows of A, and with vertex i joinedto vertex j (j 6= i) if aij 6= 0. We call the matrix A connected if this underlyinggraph is connected. If the matrix is not connected, then for some permutationmatrix P , the matrix PTAP is in block diagonal form, with blocks corresponding toconnected components of the underlying graph: we call these blocks the componentsof A. We write tr(A) for the trace of A. The eigenvalues of A are the roots of itscharacteristic polynomial, pA(x) = det(xI−A). These are all real, since A is a realsymmetric matrix (with integer entries).The product of the eigenvalues of A is an integer. If all the eigenvalues of Aare strictly positive, then this integer is at least 1, and the inequality of arithmeticand geometric means implies that tr(A) ≥ n. Our main theorem is a best-possiblesharpening of this inequality.Theorem 1. Let A be an n × n, connected, integer symmetric matrix. If all theeigenvalues of A are strictly positive, then tr(A) ≥ 2n− 1.Although we require A to be connected in the statement of the theorem, we donot require that the characteristic polynomial pA(x) is irreducible. On the otherhand, if pA(x) is irreducible, then A is certainly connected.The absolute trace of a totally positive algebraic integer (one whose galois con-jugates are all real and strictly positive) is defined to be its trace divided by itsdegree. The Schur-Siegel-Smyth trace problem (so-called in [3]; see [2] for a survey)asks if 2 is the smallest limit point of the set of absolute traces of totally positivealgebraic integers. It is known that the smallest limit point is at least 1.79193 ([8];and see [13, 14, 15, 7, 11, 1, 2, 6, 9] for earlier records).1991 Mathematics Subject Classification. 15A42.Key words and phrases. integer symmetric matrices; trace problem; minimal polynomials.12 JAMES MCKEE AND PAVLO YATSYNABy analogy, one can define the absolute trace of an n×n integer symmetric matrixA to be tr(A)/n. In the case where the characteristic polynomial is irreducible, andall eigenvalues are strictly positive, this is just the absolute trace of any one of theeigenvalues. The conclusion of Theorem 1 is equivalent to the assertion that A hasabsolute trace at least 2− 1/n. We then easily get the following corollary.Corollary 2. Let X be the set of absolute traces of connected integer symmetricmatrices having all their eigenvalues strictly positive. Then the smallest limit pointof X is 2. Thus the integer symmetric matrix analogue of the Schur-Siegel-Smythtrace problem is settled.Estes and Guralnick [5] considered the question of which separable monic to-tally real integer polynomials are minimal polynomials of integer symmetric ma-trices. They showed that for degree up to 4, all such polynomials arise as minimalpolynomials, and conjectured that this might be true for all larger degrees too.Dobrowolski [4] showed that if the discriminant of a polynomial is too small (com-pared to a function of its degree), then it cannot be the minimal polynomial of aninteger symmetric matrix. He showed the existence of infinitely many counterex-amples, derived from cyclotomic polynomials. The smallest degree of any of thesecounterexamples is 2880. The first author [9] gave a classification of all integersymmetric matrices whose span (the difference between the largest and smallesteigenvalues) is strictly less than 4. This led to the discovery of some counterexam-ples of much lower degree, including three examples of degree 6. Theorem 1 showsthat if the trace of an irreducible polynomial is too small (compared to a function ofits degree), then it cannot be the characteristic polynomial of an integer symmetricmatrix. It is not hard to deduce the following corollary, which can be used to givemore counterexamples to the Estes-Guralnick conjecture.Corollary 3. Let m be a monic, irreducible integer polynomial of degree n, withall roots real and strictly positive. If the trace of m is strictly less than 2n− 1, thenm is not the minimal polynomial of an integer symmetric matrix.The smallest degree to which this corollary can usefully be applied is 10. Thethree degree 10, trace 18 polynomialsx10 − 18x9 + 134x8 − 537x7 + 1265x6 − 1798x5 + 1526x4 − 743x3 + 194x2 − 24x+ 1 ,x10 − 18x9 + 134x8 − 538x7 + 1273x6 − 1822x5 + 1560x4 − 766x3 + 200x2 − 24x+ 1 ,x10 − 18x9 + 135x8 − 549x7 + 1320x6 − 1920x5 + 1662x4 − 813x3 + 206x2 − 24x+ 1 ,which appeared first in [10], give counterexamples to the Estes-Guralnick conjecturethat do not yield to the methods of [4] or [9] (the discriminants and spans are toolarge). For all d, there exists n such that there is a totally positive irreduciblepolynomial of degree n and trace 2n−d: see [11]. The smallest degree n known forwhich the trace is as small as 2n− 3 is the degree-27, trace-51 example given in [9].2. Proofs2.1. Proof of Theorem 1. Suppose that the theorem is false. Let A = (aij) bean n×n counterexample with n minimal, and with the trace minimal for this valueof n. In particular, A is symmetric, connected, has all eigenvalues strictly positive,and tr(A) ≤ 2n− 2.Let e1, . . . , en be a basis for Rn. Having chosen a basis, the matrix A defines asymmetric bilinear form 〈·, ·〉 on Rn ×Rn via 〈ei, ej〉 = aij . Since all eigenvalues ofTRACE BOUND 3A are strictly positive, this bilinear form is positive definite, and hence aii > 0 foreach i.As A is supposed to be a counterexample, tr(A) ≤ 2n − 2. With the diagonalentries being strictly positive integers, at least two of them must equal 1. (Inparticular, n ≥ 2.) Shuﬄing rows and columns (i.e., relabelling the basis vectors),we may assume that 〈e1, e1〉 = 1.Being a counterexample, the underlying graph of A is connected, so we must have〈e1, ej〉 6= 0 for some j > 1: relabelling we may suppose that 〈e1, e2〉 6= 0. (Here weuse n ≥ 2. Were it not for this detail, the proof would work for tr(A) = 2n− 1.)Let e′2 = e2 − a12e1. Then e1, e′2, e3, . . . , en is another basis of Rn, and thematrix A′ = (a′ij) of our bilinear form with respect to this new basis still hasinteger entries, and all eigenvalues are strictly positive. We have a′12 = a′21 = 0,and a′22 = a22 − a212. The matrices A and A′ agree in all rows and columns exceptthe second.The trace of A′ is strictly smaller than that of A, since a′22 < a22, so by the mini-mality assumptions on our counterexample, we must have that A′ is not connected.We now show that A′ has exactly two components.Take any j between 3 and n, and let x1, . . . , xr be a path from 2 to j in theunderlying graph of A (so x1 = 2, xr = j, and for 1 ≤ i < r we have axixi+1 6= 0,and the xi are distinct; a walk is similar to a path, but without the requirement ofhaving distinct vertices). If this path remains a path in the underlying graph of A′,then j is in the same component as 2 in A′. Otherwise, a2x2 6= 0, but a′2x2 = 0 (thisfirst edge is the only edge in the path that can have been deleted in moving fromA to A′, as it is the only one incident with vertex 2). Since a′2x2 = a2x2 − a12a1x2 ,we must have a1x2 6= 0. Replacing x1 = 2 by x1 = 1, we get a walk from 1 to j inthe underlying graph of A′, so j is in the same component as 1 in A′. Thus A′ hasat most (and hence exactly) two components: the component containing 1 and thecomponent containing 2.Let the components of A′ have r1 and r2 rows, so that r1 + r2 = n. Let thetraces of these components be t1 and t2. Since t1 + t2 = tr(A′) < tr(A) < 2n − 1,we have t1 + t2 ≤ 2(r1 + r2)− 3, and hence either t1 < 2r1 − 1 or t2 < 2r2 − 1, orboth. So at least one of the two components would give a smaller counterexample,giving a contradiction.Hence the theorem cannot be false.2.2. Proof of Corollary 2. The tridiagonal n× n matrix1 1 0 · · · 01 2 1 · · · 00 1 2 · · · 0.......... . ....0 0 0 · · · 2is totally positive (one way to see this is to pick an orthonormal basis e1, . . . , enfor Rn, and note that the above matrix is the Gram matrix for the basis e1, e1+e2,e2+e3, . . . , en−1+en) and has trace 2n−1. Connectedness is plain: the underlyinggraph is a path. Hence 2 is a limit point of the set of absolute traces of connectedtotally positive integer symmetric matrices.Take any  > 0. By Theorem 1, any connected totally positive n × n integersymmetric matrix of absolute trace below 2 −  has n ≤ 1/, so there are finitely4 JAMES MCKEE AND PAVLO YATSYNAmany possibilities for n. For each n, the bound on the absolute trace gives a finiteset of possibilities for the diagonal entries of the matrix. Then since for any i < jthe submatrix (aii aijaij ajj)must be positive definite, the off-diagonal aij are bounded. Hence, for any  > 0,the number of connected totally positive n × n integer matrices of absolute tracebelow 2−  is finite, and the corollary follows.2.3. Proof of Corollary 3. Let m be a totally positive irreducible monic integerpolynomial of degree n and trace at most 2n − 2. Suppose that m is the minimalpolynomial of an integer symmetric matrix. Then the characteristic polynomialof that matrix is a power of m. If the matrix were not connected, then eachcomponent would have m as its minimal polynomial. Hence the smallest integersymmetric matrix A for which m is the minimal polynomial would be connected.Then A would be rn × rn for some r, with characteristic polynomial mr havingtrace at most 2nr − 2r < 2nr − 1. This would contradict Theorem 1References[1] J. Aguirre, M. Bilbao, J.C Peral, The trace of totally positive algebraic integers, Math. Comp.75 (2006), no. 253, 385–393.[2] J. Aguirre, J.C Peral,The trace problem for totally positive algebraic integers, Number Theoryand Polynomials, London Mathematical Society Lecture Note Series 352, 1–19, CambridgeUniversity Press, Cambridge 2008.[3] P. Borwein, Computational excursions in analysis and number theory, CMS Books in Math-ematics, 10, Springer-Verlag, New York 2002.[4] E. Dobrowolski, A note on integer symmetric matrices and Mahler’s measure,Canad. Math. Bull. 51 (2008), 57–59.[5] D.R. Estes, R.M. Guralnick, Minimal Polynomials of Integral Symmetric Matrices, LinearAlgebra and its Applications 192 (1993), 83–99.[6] V. Flammang, Trace of totally positive algebraic integers and integer transfinite diameter,Math. Comp. 78 (2009), no. 266, 1199–1125.[7] V. Flammang, M. Grandcolas, G. Rhin, Small Salem numbers, Number theory in progress,Vol. 1 (Zakopane-Kos´cielisko 1997), 165–168, de Gruyter, Berlin, 1999.[8] Y. Liang, Q. Wu, The trace problem for totally positive algebraic integers, J. Aust. Math. Soc.90 (2011), 341–354.[9] J. McKee, Computing totally positive algebraic integers of small trace, Math. Comp. 80(2011), no. 274, 1041–1052.[10] J. McKee, C.J. Smyth, Salem numbers of trace −2 and traces of totally positive algebraicintegers, Algorithmic number theory, Lecture Notes in Computer Science 3076, 327–337,Springer Verlag, Berlin, 2004.[11] , There are Salem numbers of every trace, Bull. London Math. Soc 37 (2005), 25–36.[12] I. Schur, U¨ber die Verteilung der Wurzeln bei gewissen algebraischen Gleichungen mit ganz-zahligen Koeffizienten, Math. Z. 1 (1918), no. 4, 377–402.[13] C.L. Siegel, The trace of totally positive and real algebraic integers, Ann. of Math. (2) 46(1945), 302–312.[14] C.J. Smyth, Totally positive algebraic integers of small trace, Ann. Inst. Fourier (Grenoble)34 (1984), no. 3, 1–28.[15] , The mean values of totally real algebraic integers, Math. Comp. 42 (1984), no. 166,663–681.Department of Mathematics, Royal Holloway, University of London, Egham Hill,Egham, Surrey, TW20 0EX, England, UK.E-mail address: james.mckee@rhul.ac.uk, pavlo.yatsyna.2012@live.rhul.ac.uk

James Mckee

CiteSeerX

A TRACE BOUND FOR POSITIVE DEFINITE CONNECTED INTEGER SYMMETRIC MATRICES

https://pure.royalholloway.ac.uk/portal/files/18401096/ISMtrace.pdf

A trace bound for positive definite connected integer symmetric matrices

Abstract

Similar works

Full text

Available Versions

Crossref

Royal Holloway - Pure

CiteSeerX