NON-ARCHIMEDEAN VALUATIONS OF EIGENVALUES OF MATRIX POLYNOMIALS MARIANNEAKIAN,RAVINDRABAPAT,ANDSTE´PHANEGAUBERT 6 1 0 2 Abstract. We establish general weak majorization inequalities, relating the n leadingexponentsoftheeigenvaluesofmatricesormatrixpolynomialsoverthe a fieldofPuiseuxserieswiththetropicalanaloguesofeigenvalues. Wealsoshow J thattheseinequalitiesbecomeequalitiesundergenericityconditions,andthat 4 theleadingcoefficientsoftheeigenvaluesaredeterminedastheeigenvaluesof auxiliarymatrixpolynomials. ] P 1. Introduction S . 1.1. Non-archimedean valuations and tropical geometry. A non- h t archimedean valuation ν on a field K is a map K → R ∪ {+∞} such that a m (1a) ν(a)=+∞ ⇐⇒ a=0 [ (1b) ν(a+b)(cid:62)min(ν(a),ν(b)) 1 v (1c) ν(ab)=ν(a)+ν(b) . 8 3 These properties imply that ν(a + b) = min(ν(a),ν(b)) for a,b ∈ K such that 4 ν(a) (cid:54)= ν(b). Therefore, the map ν is almost a morphism from K to the min- 0 plus or tropical semifield R , which is the set R ∪ {+∞}, equipped with the min 0 addition (a,b) (cid:55)→ min(a,b) and the multiplication (a,b) (cid:55)→ a + b. A basic ex- . 1 ample of field with a non-archimedean valuation is the field of complex Puiseux 0 series, with the valuation which takes the leading (smallest) exponent of a series. 6 The images by a non-archimedean valuation of algebraic subsets of Kn are known 1 as non-archimedean amoebas. The latter have a combinatorial structure which is : v studied in tropical geometry [IMS07, MS15]. For instance, Kapranov’s theorem i X showsthattheclosureoftheimagebyanon-archimedeanvaluationofanalgebraic hypersurface over an algebraically closed field is a tropical hypersurface, i.e., the r a non-differentiability locus of a convex polyhedral function, see [EKL06]. This gen- eralizes the characterization of the leading exponents of the different branches of an algebraic curve in terms of the slopes of the Newton polygon, which is part of the classical Newton-Puiseux theorem. 1.2. Main results. In the present paper, we consider the eigenproblem over the field of complex Puiseux series and related fields of functions. Our aim is to relate the images of the eigenvalues by the non-archimedean valuation with certain easily computable combinatorial objects called tropical eigenvalues. 2000 Mathematics Subject Classification. 47A55,47A75,05C50,12K10. Key words and phrases. Perturbation theory, max-plus algebra, tropical semifield, spectral theory,Newton-Puiseuxtheorem,amoeba,majorization,graphs,optimalassignment. 1 2 MARIANNEAKIAN,RAVINDRABAPAT,ANDSTE´PHANEGAUBERT Thefirstmainresultofthepresentpaper,Theorem4.4,showsthatthesequence ofvaluationsoftheeigenvaluesofamatrixA∈Kn×nisweakly(super)majorizedby the sequence of (algebraic) tropical eigenvalues of the matrix obtained by applying the valuation to the entries of A. Next, we show that the same majorization inequality holds under more general circumstances. In particular, we consider in Theorem 5.2 and Corollary 5.3 a relaxed definition of the valuation, in the spirit of large deviations theory, assuming that the entries of the matrix are functions of a small parameter (cid:15). We do not require these functions to have a Puiseux series type expansion, but assume that they have some mild form of first order asymptotics. Moreover, the results apply to a lower bound of the valuation of the entries of A. Then, in Section 7, we assume that the entries of A satisfy A =a (cid:15)Aij +o((cid:15)Aij) , ij ij for some scalars a ∈ C and A ∈ R∪{+∞}, as (cid:15) tends to 0. When a = 0, ij ij ij this reduces to Aij =o((cid:15)Aij), so that valuations are partially known: only a lower bound is known. Applying Corollary 5.3, majorization inqualities are derived in Theorem 7.1. The assumption of Section 7 is satisfied of course if the entries of A are absolutely converging Puiseux series, or more generally, if these entries belongtoapolynomiallyboundedo-minimalstructure[vdD99,Ale13]. Weshowin Theorem 7.4 that the majorization inequalities of Theorem 7.1 become equalities for generic values of the entries a . The proof of the latter theorem relies on ij some variations of the Newton-Puiseux theorem, which we state as Theorems 6.1 and6.2. Thelatterresultsonlyrequireapartialinformationontheasymptoticsof thecoefficientsofthepolynomial. Theparticularcasewherethispartialinformation containsatleastthefirstorderasymptoticsofallthecoefficientsofthepolynomial was considered in [Die68]. However, here we show that a partial information on the first order asymptotics, giving an outer approximation of a Newton polytope, allow one to derive a partial information on the roots. The latter idea goes back at least to [Mon34] in the context of archimedean valuations. The valuation only gives an information on the leading exponent of Puiseux series. Our aim in Section 8 is to refine this information, by characterizing also the coefficients λ ∈ C of the leading monomials of the asymptotic expansions of the i different eigenvalues Li of the matrix A, Li ∼λ (cid:15)Λi, 1(cid:54)i(cid:54)n . i As a byproduct, we shall end up with an explicit form, easily checkable, of the genericity conditions under which the majorization inequalities become equalities. To this end, it is necessary to embed the standard eigenproblem in the wider class ofmatrixpolynomialeigenproblems. Theorems8.2and8.3showinparticularthat thecoefficientsλ aretheeigenvaluesofcertainauxiliarymatrixpolynomialswhich i aredeterminedonlybytheleadingexponentsandleadingcoefficientsoftheentries of A. These polynomials are constructed from the optimal dual variables of an optimal assignment problem, arising from the evaluation of the tropical analogue of the characteristic polynomial. 1.3. Application to perturbation theory and discussion of related work. The present results apply to perturbation theory [Kat95, Bau85], and specially, to the singular case in which a matrix with multiple eigenvalues is perturbed. The lattersituationistheobjectofthetheorydevelopedbyViˇsikandLjusternik[VL60] NON-ARCHIMEDEAN VALUATIONS OF EIGENVALUES OF MATRIX POLYNOMIALS 3 and completed by Lidski˘ı [Lid65], see [MBO97] for a survey. The goal of this theory is to give a direct characterization of the exponents, without computing the Newton polytope of the characteristic polynomial. The theorem of [Lid65] solves this problem under some genericity assumptions, requiring the non-vanishing of certainSchurcomplements. ThequestionofsolvingdegenerateinstanceofLidsk˘ı’s theorem has been considered in particular, by Ma and Edelman [ME98] and Na- jman [Naj99], and also by Moro, Burke and Overton in [MBO97]. Theorems 8.2 and 8.3 generalize the theorem of Lidski˘ı, as they allow one to solve degenerate instances in which the Schur complements needed in Lidski˘ı’s construction are no longer defined. The present train of thoughts originates from a work of Friedland [Fri86], who showed that a certain deformation of the Perron root of a nonnegative matrix, in terms of Hadamard powers, converges to the maximal circuit mean of the matrix, a.k.a.,themaximaltropicaleigenvalue. Then,in[ABG98],weshowedthatthelim- iting Perron eigenvector, along the same deformation, can also be characterized by tropical means, under a nondegeneracy condition. An early version of the present Theorems 7.1 and 7.4 appeared as Theorem 3.8 of the authors’ preprint [ABG04a]. There, we also gave a generalization of the theorem of Lidski˘ı, in which the exponents of the first order asymptotics of the eigenvalues are given by the tropical eigenvalues of certain tropical Schur complements and their coefficients are given bysomeassociatedusualSchurcomplementslikeinthetrueLidski˘ıtheorem. How- ever, some singular cases remained, see for instance Example 8.4, motivating the introduction of matrix polynomial eigenproblems in further works. The results of Theorems 8.2 and 8.3 were announced without proof in the note [ABG04b]. Therefore, the present article is, for some part, a survey of results which have not appeared previously in the form of a journal article. It also provides a general presentation of eigenvalues in terms of valuation theory, with several new results or refinements, like the general majorization inequality for the eigenvalues of matrix polynomials, Theorem 8.1. The interest of this presentation is that it explains better the relation between the results obtained here, or in eigenvalues perturbation theory, for the non-archimedean valuations of matrix entries and eigenvalues, withtheiranaloguesforarchimedeanvaluations,likethemodulusmap,orforsome generalization of the notion of archimedean valuation which includes in particular matrix norms. Indeed, the latter works have motivated a more recent work by Akian, Gaubert and Marchesini, who showed in [AGM14], that one form of the theorem of Fried- land concerning the Perron or dominant root carries over to all eigenvalues: the sequence of moduli of all eigenvalues of a matrix is weakly log-majorized by the sequence of tropical eigenvalues, up to certain combinatorial coefficients. There- fore, the result there can be thought of as a analogue of Theorem 4.4 or 7.1 for the modulus archimedean valuation. Also, the results of [ABG04a, ABG04b] have been at the origin of the application of tropical methods by Gaubert and Sharify to the numerical computation and estimation of eigenvalues [GS09, Sha11], based on the norms of matrix polynomial coefficients. This is a subject of current interest, with work by Akian, Bini, Gaubert, Hammarling, Noferini, Sharify, and Tisseur [BNS13, HMT13, NST14, AGS13]. The present results provide a further illustration of the role of tropical algebra in asymptotic analysis, which was recognized by Maslov [Mas73, Ch. VIII]. 4 MARIANNEAKIAN,RAVINDRABAPAT,ANDSTE´PHANEGAUBERT He observed that WKB-type or large deviation type asymptotics lead to limiting equations, like Hamilton-Jacobi equations, of a tropical nature. This observation is at the origin of idempotent analysis [MS92, DKM92, KM97, LMS01]. The same deformation has been identified by Viro [Vir01], in relation with the patchworking method he developed for real algebraic curves. Note that all the perturbation results described or recalled above study suffi- cient conditions for the possible computation of first order asymptotics of some roots when an information on the first order asymptotics of the data is only avail- able. However,whenallthePuiseuxseriesexpansionofthedataisknown,Puiseux theoremallowsonetocomputewithoutanyconditionallthePuiseuxseriesexpan- sion. In the context of matrix polynomials, Murota [Mur90], gave an algorithm to compute the Puiseux series expansions of the eigenvalues of a matrix polynomial depending polynomially in the parameter (cid:15), avoiding the explicit computation of the characteristic polynomial. As for Theorems 8.2 and 8.3, his algorithm relies on a parametric optimal assignment problem. The present work builds on tropical spectral theory. It has been inspired by the analogy with nonnegative matrix theory, of which Hans Schneider was a master. We gratefully acknowledge our debt to him. 2. Min-plus polynomials and Newton polygons Wefirstrecallsomeelementaryfactsconcerningformalpolynomialsandpolyno- mialfunctionsoverthemin-plussemifield,andtheirrelationwithNewtonpolygons. The min-plus semifield, R , is the set R∪{+∞} equipped with the addition min (a,b) (cid:55)→ min(a,b), denoted a⊕b, and the multiplication (a,b) (cid:55)→ a+b, denoted a⊗b or ab. We shall denote by 0 = +∞ and 1 = 0 the zero and unit elements of R , respectively. The familiar algebraic constructions and conventions carry out min to the min-plus context with obvious changes. For instance, if A,B are matrices of compatible dimensions with entries in R , we shall denote by AB the matrix min product with entries (AB) = (cid:76) A B = min (A +B ), we denote by Ak ij k ik kj k ik kj the kth min-plus matrix power of A, etc. Moreover, if x∈R \{0}, then we will min denote by x−1 the inverse of x for the ⊗ law, which is nothing but −x, with the conventional notation. The reader seeking information on the min-plus semifield may consult [CG79, MS92, BCOQ92, KM97, ABG13, But10]. We denote by R [Y] the semiring of formal polynomials with coefficients in min R in the indeterminate Y: a formal polynomial P ∈ R [Y] is nothing but a min min sequence (Pk)k∈N ∈ RNmin such that Pk = 0 for all but finitely many values of k. Formal polynomials are equipped with the entry-wise sum, (P ⊕Q) = P ⊕Q , k k k (cid:76) and the Cauchy product, (PQ) = P Q . As usual, we denote a formal k 0(cid:54)i(cid:54)k i k−i polynomial P as a formal sum, P = (cid:76)∞ P Yk. We also define the degree and k=0 k valuation of P: degP = sup{k ∈ N | P (cid:54)= 0}, valP = inf{k ∈ N | P (cid:54)= 0} k k (degP = −∞ and valP = +∞ if P = 0). To any P ∈ R [Y], we associate the min polynomial function P(cid:98) : Rmin → Rmin, y (cid:55)→ P(cid:98)(y) = (cid:76)∞k=0Pkyk, that is, with the usual notation: (2) P(cid:98)(y)=min(Pk+ky) . k∈N Thus,P(cid:98) isconcave,piecewiseaffinewithintegerslopes. WedenotebyRmin{Y}the semiring of polynomial functions P(cid:98). The morphism Rmin[Y] → Rmin{Y}, P (cid:55)→ P(cid:98) NON-ARCHIMEDEAN VALUATIONS OF EIGENVALUES OF MATRIX POLYNOMIALS 5 isnotinjective,asitisessentiallyaspecializationoftheclassicalFencheltransform over R, which reads: R R F :R →R , F(g)(y)=sup(xy−g(x)) , x∈R where R := R∪{±∞}. Indeed, for all y ∈ R, P(cid:98)(y) = −F(P)(−y), where the function k (cid:55)→P from N to R is extended to a function k min (cid:40) P if x=k ∈N , (3) P :R→R , x(cid:55)→P(x), with P(x)= k min +∞ otherwise ThefollowingresultofCuninghame-GreenandMeijergivesamin-plusanalogue of the fundamental theorem of algebra. Theorem 2.1 ([CGM80]). Any polynomial function P(cid:98) ∈Rmin{Y} can be factored in a unique way as (4) P(cid:98)(y)=Pn(y⊕c1)···(y⊕cn) , with c (cid:54)···(cid:54)c . 1 n The ci will be called the roots of P(cid:98). The multiplicity of the root c is the cardinality of the set {j ∈ {1,...,n} | cj = c}. We shall denote by R(P(cid:98)) the sequence of roots: R(P(cid:98)) = (c1,...,cn). By extension, if P ∈ Rmin[Y] is a formal polynomial, we will call roots of P the roots of P(cid:98), so R(P) := R(P(cid:98)). The next properties also follow from [CGM80]; they show that the definition of the roots in Theorem2.1isaspecialcaseofthenotionofatropicalhypersurfacedefinedasthe nondifferentiability locus of a tropical polynomial [IMS07]. Proposition 2.2 (See [CGM80]). The roots c ∈ R of a formal polynomial P ∈ Rmin[Y] are exactly the points at which the function P(cid:98) is not differentiable. The multiplicity of a root c ∈ R is equal to the variation of slope of P(cid:98) at c, P(cid:98)(cid:48)(c−)− P(cid:98)(cid:48)(c+). Moreover, 0 is a root of P if, and only if, P(cid:98)(cid:48)(0−) := limc→+∞P(cid:98)(cid:48)(c) (cid:54)= 0. In that case P(cid:98)(cid:48)(0−) is the multiplicity of 0, and it coincides with valP. (cid:3) Legendre-Fenchel duality allows one to relate the tropical roots to the slopes of Newtonpolygons. Toseethis,denotebyvexf theconvexhullofamapf :R→R, and denote by P the formal polynomial whose sequence of coefficients is obtained by restricting to N the map vexP: k (cid:55)→P :=(vexP)(k), for k ∈N. The function k vexP isfiniteontheinterval[valP,degP]. Also,itshouldbenotedthatthegraph ofvexP isthestandardNewtonpolygonassociatedtothesequenceofpoints(k,P ), k k ∈[valP,degP]. Theorem 2.3 ([BCOQ92, Th. 3.43, 1 and 2]). A formal polynomial of degree n, P ∈R [Y], satisfies P =P if, and only if, there exist c (cid:54)···(cid:54)c ∈R such min 1 n min that P =P (Y⊕c )···(Y⊕c ) . n 1 n The c are unique and given, by: i (cid:40) P (P )−1 if P (cid:54)=0 (5) c = n−i n−i+1 n−i+1 for i=1,...,n . i 0 otherwise, The following standard observation relates the tropical roots with the Newton polygon. 6 MARIANNEAKIAN,RAVINDRABAPAT,ANDSTE´PHANEGAUBERT Proposition 2.4. The roots c ∈ R of a formal polynomial P ∈ R [Y] coincide min with the opposites of the slopes of the affine parts of vexP :[valP,degP]→R. The multiplicity of a root c∈R coincides with the length of the interval in which vexP has slope −c. (cid:3) Remark 2.5. The duality between tropical roots and slopes of the Newton polygon in Proposition 2.4 is a special case of the Legendre-Fenchel duality formula for subdifferentials: −c ∈ ∂(vexP)(x) ⇔ x ∈ ∂F(P)(−c) ⇔ x ∈ ∂+P(cid:98)(c) where ∂ and ∂+ denote the subdifferential and superdifferential, respectively [Roc70, Th. 23.5]. TheabovenotionsareillustratedinFigure1,whereweconsidertheformalmin- plus polynomial P =Y3⊕5Y2⊕6Y⊕13. The map j (cid:55)→P , together with the map j vexP, are depicted at the left of the figure, whereas the polynomial function P(cid:98) is depictedattherightofthefigure. WehaveP =Y3⊕3Y2⊕6Y⊕13=(Y⊕3)2(Y⊕7). Thus, the roots of P are 3 and 7, with respective multiplicities 2 and 1. The roots arevisualizedattherightofthefigure,oralternatively,astheoppositeoftheslopes of the two line segments at the left of the figure. The multiplicities can be read either on the map P(cid:98) at the right of the figure (the variation of slope of P(cid:98) at points 3 and 7 is 2 and 1, respectively), or on the map vexP at the left of the figure (as the respective horizontal widths of the two segments). We conclude this section by 13 =P 13 =P¯ 5+2y 6+y 9 =vexP 9 6 5 5 3y 0 0 0 1 2 3 0 3 7 Figure 1. TheNewtonpoygonoftheformalmin-pluspolynomial P =Y3⊕5Y2⊕6Y⊕13(left)andtheassociatedpolynomialfunction P(cid:98) (right). two technical results. Lemma 2.6. Let P = (cid:76)n P Yi ∈ R [Y] be a formal polynomial of degree n. i=0 i min Then, R(P)=(c (cid:54)···(cid:54)c ) if, and only if, P (cid:62)P (Y⊕c )···(Y⊕c ) and 1 n n 1 n (6) P =P c ···c for all i∈{0,n}∪{i∈{1,...,n−1}| c 0. n n k k 0 0 0 0 We have shown (6), together with the last statement of the lemma. Since P =P n n and P (cid:62)P, we also obtain P (cid:62)P (Y⊕c )···(Y⊕c ). n 1 n For the “if” part, assume that P (cid:62) P (Y⊕c )···(Y⊕c ) and that (6) holds. n 1 n Since Q = P (Y⊕c )···(Y⊕c ) is convex, and the convex hull map P (cid:55)→ P is n 1 n monotone, we must have P (cid:62) Q = Q. Hence, P (cid:62) P (cid:62) Q and since P = Q n−i n−i foralliasin(6),wemusthaveP =P =Q ,thusvexP(n−i)=Q(n−i)at n−i n−i n−i thesei. SincevexP andQareconvex,QispiecewiseaffineandQ(j)=vexP(j)for j attheboundaryofthedomainofQandatallthej whereQchangesofslope,we must have vexP =Q. Hence P =Q and R(P)=R(P)=R(Q)=(c ,...,c ). (cid:3) 1 n Corollary 2.7. Let P =(cid:76)n P Yi ∈R [Y] be a formal polynomial of degree n. i=0 i min Let c∈R be a finite root of P with multiplicity m, and denote by m(cid:48) the sum of the multiplicities of all the roots of P greater than c (+∞ comprised). Then, P = P i i for both i = m(cid:48) and i = m+m(cid:48), P(cid:98)(c) = Pm(cid:48)cm(cid:48) = Pm+m(cid:48)cm+m(cid:48) and P(cid:98)(c) < Pici for all 1(cid:54)i 0, and c < c if n−m(cid:48) < n. By Lemma 6, this implies n−m(cid:48)+1 that for both i = m(cid:48) and i = m+m(cid:48), P = P = P c ···c . We also have i i n 1 n−i Since P = (vexP)(i) (cid:54) P , we have P (cid:62) (vexP)(i) = P = P c ···c for i i i i n 1 n−i all i = 0,...,n. Moreover, by Theorem 2.1, we have P(cid:98)(c) = Pn(c⊕c1)···(c⊕ cn)=Pnc1···cn−m(cid:48)−mcm+m(cid:48) =Pnc1···cn−m(cid:48)cm(cid:48). Hence,P(cid:98)(c)=Pm+m(cid:48)cm+m(cid:48) = Pm(cid:48)cm(cid:48), and P(cid:98)(c)m+m(cid:48). (cid:3) 3. Tropical eigenvalues We now recall some classical results on tropical eigenvalues and characteristic polynomials. Thepermanentofamatrixwithcoefficientsinanarbitrarysemiring(S,⊕,⊗)is defined by n (cid:77) (cid:79) per(A)= A , iσ(i) σ∈Sn i=1 where S is the set of permutations of [n] := {1,...,n}. In particular, for any n matrix A∈Rn×n, min per(A)= min |σ| , A σ∈Sn where for any permutation σ ∈S , we define the weight of σ with respect to A as n |σ| :=A +···+A . A 1σ(1) nσ(n) To any min-plus n×n matrix A, we associate the (directed) graph G(A), which has set of nodes [n] and an arc (i,j) if A (cid:54)= 0, and the weight function which ij 8 MARIANNEAKIAN,RAVINDRABAPAT,ANDSTE´PHANEGAUBERT associates the weight A to the arc (i,j) of G(A). In the sequel, we shall omit ij the word “directed” as all graphs will be directed. Then, per(A) is the value of an optimal assignment in this weighted graph. It can be computed in O(n3) time using the Hungarian algorithm [Sch03, § 17]. We refer the reader to [BR97, § 2.4] or [Sch03, § 17] for more background on the optimal assignment problem and a discussion of alternative algorithms. We define the formal characteristic polynomial of A, n (cid:77) (cid:79) P :=per(YI⊕A)= (Yδ ⊕A )∈R [Y] , A iσ(i) iσ(i) min σ∈Sn i=1 whereI istheidentitymatrix,andδ =1ifi=jandδ =0otherwise. Theformal ij ij polynomial P has degree n and its coefficients are given by (P ) =trmin(A), for A A k n−k k =0,...,n−1 and (P ) =1, where trmin(A) is the min-plus k-th trace of A: A n k   (cid:77) (cid:77) (cid:79) (7) trmkin(A):=  Ajσ(j) , J⊂{1,...,n},#J=k σ∈SJ j∈J where S is the set of permutations of J. The associated min-plus polynomial J function will be called the characteristic polynomial function of A, and its roots will be called the (algebraic) eigenvalues of A. The algebraic eigenvalues of A (and so, its characteristic polynomial function) can be computed in O(n4) time by the method of Burkard and Butkoviˇc [BB03]. Gassner and Klinz [GK10] showed that this can be reduced to a O(n3) time, using parametric optimal assignment techniques. However, it is not known whether the sequenceofcoefficientsoftheformalcharacteristicpolynomialP canbecomputed A in polynomial time. The term algebraic eigenvalue is used here since unlike for matrices with real or complexcoefficients,arootλ∈R ofthecharacteristicpolynomialofan×nmin- min plusmatrixAmaynotsatisfyAu=λuforsomeu∈Rn . Toavoidanyconfusion, min weshallcallascalarλwiththelatterpropertyageometric eigenvalue. Thefollow- ing statement and remarks collect some results in tropical spectral theory, which have been developed by several authors [CG79, Vor67, Rom67, GM77, CDQV83, MS92, BSvdD95, AGW05, BCGG09]. We refer the reader to [BCOQ92, But10] for more information. We say that a matrix A is irreducible if G(A) is strongly connected. Theorem 3.1 (See e.g. [But10]). The minimal algebraic eigenvalue of a matrix A∈Rn×n is given by min n (8) ρmin(A)=(cid:77) (cid:77) (Ai1i2···Aiki1)k1 , k=1i1,...,ik or equivalently, by the following expression called minimal circuit mean, |c| (9) min A , ccircuitinG(A) |c| where for all paths p = (i ,i ,...,i ) in G(A), we denote by |p| = A +···+ 0 1 k A i0i1 A the weight of p, and by |p| = k its length, and the minimum is taken over ik−1ik all elementary circuits of G(A). (cid:3) NON-ARCHIMEDEAN VALUATIONS OF EIGENVALUES OF MATRIX POLYNOMIALS 9 An important notion to be used in the sequel is the one of critical circuit, i.e., of circuit c=(i ,i ,...,i ,i ) of G(A) attaining the minimum in (9). The critical 1 2 k 1 graph of A is the union of the critical circuits, that is the graph whose nodes and arcs belong to critical circuits. It is known that ρ (A) is the minimal geometric min eigenvalueofAandthatthemultiplicityotρ (A)asageometriceigenvalue(i.e., min the“dimension”oftheassociatedeigenspace)concideswiththenumberofstrongly connected components of the critical graph. Note also that, if A is irreducible, ρ (A) is the unique geometric eigenvalue of A. min Remark 3.2. Themultiplicityofρ (A),asanalgebraiceigenvalue,coincideswith min the term rank (i.e., the maximal number of nodes of a disjoint union of circuits) of the critical graph of A. This follows from the arguments of proof of Theorem 4.7 in [ABG04a]. 4. Majorization inequalities for valuations of eigenvalues The inequalities that we shall establish involve the notion of weak majorization, see [MO79] for background. Definition 4.1. Let u,v ∈ Rn . Let u (cid:54) ··· (cid:54) u (resp. v (cid:54) ··· (cid:54) v ) min (1) (n) (1) (n) denote the components of u (resp. v) in increasing order. We say that u is weakly (super) majorized by v, and we write u≺w v, if the following conditions hold: u ···u (cid:62)v ···v ∀k =1,...,n . (1) (k) (1) (k) Theweakmajorizationrelationisonlydefinedin[MO79]forvectorsofRn. Here, it is convenient to define this notion for vectors with infinite entries. We used the min-plusnotationforhomogeneitywiththerestofthepaper. Thefollowinglemma statesausefulmonotonicitypropertyofthemapwhichassociatestoaformalmin- plus polynomial P its sequence of roots, R(P). Lemma 4.2. Let P,Q∈R [X] be two formal polynomials of degree n. Then, min (10) P (cid:62)Q and P =Q =⇒ R(P)≺w R(Q) . n n Proof. From P (cid:62) Q, we deduce P (cid:62) Q. Let R(P) = (c (P) (cid:54) ··· (cid:54) c (P)) and 1 n R(Q) = (c (Q) (cid:54) ··· (cid:54) c (Q)) denote the sequence of roots of P and Q, respec- 1 n tively. Using P (cid:62) Q, P = P = Q = Q and (5), we get c (P)···c (P) = n n n n 1 k P (P )−1 (cid:62) Q (Q )−1 = c (Q)···c (Q), for all k = 1,...,n, that is n−k n n−k n 1 k R(P)≺w R(Q). (cid:3) Let ν be a (non-archimedean) valuation on a field K, i.e., a map ν : K → R∪{+∞}satisfyingtheconditions(1)recalledintheintroduction. Weshallthink of the images of ν as elements of the min-plus semifield, writing ν(ab)=ν(a)ν(b). The main example of valuation considered here is obtained by considering the field of complex Puiseux series, with the valuation which takes the smallest exponent of a series. Recall that this field consists of the series of the form (cid:80)∞ a (cid:15)k/s with k=K k a ∈ C, K ∈ Z and s ∈ N\{0}, in which case the smallest exponent is equal to k K/s as soon as a (cid:54)= 0. The results of the present paper apply as well to formal K series or to series that are absolutely convergent for a sufficiently small positive (cid:15). The following proposition formulates in terms of tropical roots a well known propertyusuallystatedintermsofNewtonpolygons,seeforinstance[Bou89,Exer. VI.4.11]. It is a special case of a result proved in [EKL06] for non-archimedean 10 MARIANNEAKIAN,RAVINDRABAPAT,ANDSTE´PHANEGAUBERT amoebas of hypersurfaces. We include a proof relying on Lemma 2.6 for the con- venience of the reader, since we shall use the same argument in the sequel. Proposition 4.3 (See [EKL06, Th. 2.1.1]). Let K be an algebraically closed field witha(non-archimedean)valuationν andletP =(cid:80)n P Yk ∈K[Y],withP =1. k=0 k n Then, the valuations of the roots of P (counted with multiplicities) coincide with the roots of the min-plus polynomial ν(P):=(cid:76)n ν(P )Yk. k=0 k Proof. Let Y ,...,Y denote the roots of P, ordered by nondecreasing valu- 1 n ation, c := ν(Y ), so that c (cid:54) ··· (cid:54) c , Q := (cid:76)n c ...c Yn−k, and i i 1 n k=0 1 k P := ν(P). Observe that P = Q = 1, and Q = (Y ⊕c )...(Y ⊕c ). Since n n 1 n P =(−1)k(cid:80) Y ···Y , we get ν(P )(cid:62)ν(Y ···Y )=c ···c , and son−Pk(cid:62) Q, and Pi1<=···

Non-archimedean valuations of eigenvalues of matrix polynomials

Similar Non-archimedean valuations of eigenvalues of matrix polynomials