On the Davenport-Mahler bound

Transcription

1 On the Davenpot-Mahle boun Paula Escocielo Daniel Peucci Depatamento e Matemática, FCEN, Univesia e Buenos Aies, Agentina IMAS, CONICET UBA, Agentina Decembe 22, 206 Abstact We pove that the Davenpot-Mahle boun hols fo abitay gaphs with vetices on the set of oots of a given univaiate polynomial with complex coefficients. Intouction The Davenpot-Mahle boun is a lowe boun fo the pouct of the lengths of the eges on a gaph whose vetices ae the complex oots of a given univaiate polynomial P C[X], une cetain assumptions. Its oigins ae the wok of Mahle [0], whee a lowe boun fo the minimum sepaation between two oots of P in tems of the isciminant of P is given, an the wok of Davenpot see [2, Poposition 8], whee fo the fist time a lowe boun fo the joint pouct of many iffeent istances between oots of P which is not simply the pouct of a lowe boun fo each istance is obtaine. Roughly speaking, this boun makes evient an inteaction between the involve istances, in the sense that if some of them ae vey small, the est cannot be that small. Thoughout the liteatue, thee ae iffeent vesions of this boun. We inclue hee the one fom [5, Theoem.] see also [7, 2]. Fist, we emin the efinitions of isciminant an Mahle measue see also [, ]. Definition Let P C[X], P X a i X v i, the isciminant of P is DiscP a 2 2 v i v j 2. Definition 2 Let P C[X], P X a i X v i, the Mahle measue of P is MP a i<j max{, v i }. i Patially suppote by the Agentinian gants UBACYT an PIP CO CON- ICET. MSC Classification: 2D0, P5. Keywos: Davenpot-Mahle boun, Root sepaation, Subisciminants.

2 Theoem Davenpot-Mahle boun Let P C[X] be a polynomial of egee. Let G V, E be a iecte gaph whose vetices {v,..., v k } ae a subset of the oots of P such that: Then. if v i, v j E, then v i v j, 2. G is acyclic,. the in-egee of any vetex is at most. v i v j DiscP /2 MP #E /2, whee DiscP an MP ae the isciminant an the Mahle measue of P. Note that when P is not a squae-fee polynomial, the boun becomes tivial since DiscP vanishes. One way to manage this situation is to consie the squae-fee pat of P as in [6, Theoem ]. Anothe way suggeste by Eigenwillig [4, Theoem.9] is though the use of subisciminants, whose efinition we ecall below. This statement is nowaays known as the Genealize Davenpot-Mahle boun. Definition 4 Let P C[X], P X a i X v i, fo, the -subisciminant of P is sdisc P a 2 I {,,} #I v j v k 2. Theoem 5 Genealize Davenpot-Mahle boun Let P C[X] be a polynomial of egee with exactly istinct complex oots. Let G V, E be a iecte gaph whose vetices {v,..., v k } ae a subset of the oots of P such that:. if v i, v j E, then v i v j, 2. G is acyclic,. the in-egee of any vetex is at most. j,k I j<k Then v i v j sdisc P /2 MP #E /2 min{,2 2}/6. It is clea that if P is a squae-fee polynomial, then an the boun by Eigenwillig is exactly the classical Davenpot-Mahle boun. In the geneal case, as seen in [, Remak 4.6], sdisc P is the fist subisciminant of P which is iffeent fom zeo. One of the main applications of the Davenpot-Mahle boun in both its classical an genealize vesion is its use in algoithmic complexity estimation as fo instance in [, 5, 8]. It has also been use in [6] to obtain sepaation bouns fo oots of multivaiate polynomial systems. The main esult in this pape is that the Genealize Davenpot-Mahle boun hols fo abitay gaphs uniecte, no loops, no multiple eges with vetices on the set of oots of P. Moe pecisely: 2

3 Theoem 6 Let P C[X] be a polynomial of egee with exactly istinct complex oots. Let G V, E be a gaph whose vetices {v,..., v k } ae a subset of the oots of P. Then v i v j sdisc P /2 MP #E /2 min{,2 2}/6. In oe to pove Theoem 6, we evisit the classical poofs an the new ingeient is the use of ivie iffeences to manage the cases whee the assumptions in pevious fomulations o not hol; specially the one about the in-egee of any vetex being at most assumption, which is the one that cannot be satisfie by simply eiecting eges. Finally, afte poving Theoem 6, we inclue some emaks an applications. Poof of the esults Fist, we ecall the efinition of ivie iffeences. Definition 7 Fo f : C C an v,..., v n C with v i v j if i < j n, the ivie iffeence f[v,..., v n ] C is efine inuctively in n by if n an if n >. f[v ] fv f[v,..., v n ] f[v,..., v n ] f[v 2,..., v n ] v v n Fo F : C C m given by F z f z,..., f m z an v,..., v n C with v i v j if i < j n, the ivie iffeence F [v,..., v n ] is efine as F [v,..., v n ] f [v,..., v n ],..., f m [v,..., v n ] C m. The only popeties we will use concening ivie iffeences ae state in the next two lemmas. We efe the eae to [9, Chapte 6] fo futhe popeties of ivie iffeences an thei use in polynomial intepolation. Lemma 8 Fo F : C C m an v,..., v n C with v i v j if i < j n, F [v,..., v n ] is the linea combination of F v,..., F v n given by F [v,..., v n ] n n h k k h F v h. v h v k Poof: We pocee by inuction on n. Fo n the ientity is obvious. Fo the inuctive step we pocee

4 as follows: n+ k2 F v + v v k v v n+ F [v,..., v n+ ] F [v,..., v n ] F [v 2,..., v n+ ] v v n+ n n h2 k k h n+ n h n+ v h v k k k h k2 k h F v h. v h v k n F v h + v h v k k F v n+ v n+ v k Lemma 9 Fo p N 0, f : C C given by fz z p, an v,..., v n C with v i v j if i < j n, f[v,..., v n ] t,...,tn N n 0 t + +tnp n+ n j j if n p +, 0 if n p + 2. Poof: We fix p an we pocee by inuction on n. Fo n the ientity is obvious. Fo the inuctive step, we consie thee cases. Fist, if n + p +, then n p + an f[v,..., v n+ ] f[v,..., v n ] f[v 2,..., v n+ ] v v n+ v v n+ v v n+ tp n+ t,...,tn N n 0 t + +tnp n+ tp n+ t,t n+ N 2 0 t +t n+ t n j v t v t n+ v t v t n+ n+ t,...,t n+ N n 0 t + +t n+ p n j t 2,...,t n+ N n 0 t 2 + +t n+ p n+ t 2,...,tn N n 0 t 2 + +tnp n+ t t 2,...,tn N n 0 t 2 + +tnp n+ t n+ j j. n+ j2 n j2 n j2 j j j 4

5 If n + p + 2, then n p + an f[v,..., v n+ ] Finally, if n + p +, then n p + 2 an f[v,..., v n ] f[v 2,..., v n+ ] v v n+ v v n+ 0. f[v,..., v n+ ] f[v,..., v n ] f[v 2,..., v n+ ] v v n+ v v n We will also use the following lemma. Lemma 0 Fo, N 0 with, i /2 i 2 + /2 2 + /2. Poof: Fo the fist inequality, we fix N 0 an pocee by inuction on +. Fo + it is clea that the equality hols. Fo the inuctive step: i i

6 Fo the secon inequality, it can be easily seen fist that the inequality hols fo 0,, 2. Fo, since!, we have that: 2! an since the inequality hols , Finally, befoe poving ou main esult, we ecall [4, Lemma.8]. Lemma If m,..., m N an i m i, then m i min{,2 2}/. i We can now give the poof of ou main esult. Poof of Theoem 6: Let P X a j X v j m j C[X] with v i v j if i < j, m i N fo i. It is easy to see that the esult hols if, so fom now we suppose 2. Without loss of geneality, we suppose also that V {v,..., v } an that the oots of P ae numbee in such a way that v v. We give a iection to each ege in E: if e is an ege joining v i an v j with i < j, we consie e v i, v j as the oiente ege going fom v i to v j. Note that now G V, E satisfies conitions an 2 in Theoems an 5. We consie the eges in E liste by e v α, v β,..., e #E v α#e, v β#e. Finally, fo j, let j N 0 be the in-egee of the vetex v j. Note that 0 since thee is no ege finishing in v, an j fo j. As seen in [4, Poposition.7], On the othe han, whee W is the Vanemone matix sdisc P /2 a i<j W. j m j /2 i<j v i v j. v i v j et W 2 v... v v 2... v 2.. v... v C. 6

7 We consie F : C C, F z, z,..., z an efine a sequence of matices W, W,..., W in C. Fist, we efine W W. Then, fo fixe j,..., 2, once W j is efine, we only moify its j-th ow if any in oe to efine W j, as follows: we take the possibly empty sublist of eges e k,..., e kj finishing in v j an take as the j-th ow of W j the ivie iffeence F [v αk,..., v αkj, v j ] j i j l l i v αki v αkl v αki v j F v αki + j l F v j v j v αkl by Lemma 8. Note that the j-th ow of W j equals the j-th ow of W, which is F v j ; an since fo i j, αk i < βk i j, the αk i -th ow of W j equals the αk i -th ow of W, which is F v αki. Then, we have that o, equivalently, et W j j l et W j v j v αkl et W j et W j v j v αkl. In this way, we can pove by evese inuction in j that fo j,..., 2, et W et W j v βe v αe, j l e E βe j an at the en we obtain et W et W v βe v αe. e E The next step is to boun et W using Haama inequality. Fo j, keeping the notation of the above paagaphs, the j-th ow of W is F [v αk,..., v αkj, v j ] an by Lemma 9 its nom equals i j t,...,t j,t j + N j + 0 t + +t j +t j + i j j v t l αk l l v t j + j Note that fo each j i thee ae i j tems. Since fo l j we have that v αkl v j, we 2 /2. 7

8 have i j t,...,t j,t j + N j + 0 t + +t j +t j + i j j v t l αk l l i 2 v j 2i j i j j i 2 i j j /2 v t j + j 2 /2 max{, v j } j j /2 max{, v j } j /2 by Lemma 0. By Haama inequality, et W j j /2 max{, v j } j #E /2 max{, v j } j. j 4 Finally, using equations, 2,, 4 an Lemma, v i v j v βe v αe e E et W etw sdisc P /2 a max{, v j } j #E /2 j j m j /2 sdisc P /2 MP #E /2 min{,2 2}/6 as we wante to pove. We inclue below some emaks consieing cases in which the boun in Theoem 6 can be slightly impove. 8

9 Remak 2 Following the notation in Theoem 6, fo j let j be the total egee of vetex v j an let min{ j j }. If P is a monic polynomial then v i v j sdisc P /2 MP #E 2 /2 min{,2 2}/6. Inee, taking into account that v αe v βe fo evey e E, we change the last pat of the poof of Theoem 6 as follows: v i v j v βe v αe e E et W etw sdisc P /2 max{, v j } max{, v j αe } /2 /2 #E /2 max{, v j e E βe } /2 m j j sdisc P /2 max{, v j } 2 j #E /2 min{,2 2}/6 j sdisc P /2 MP #E 2 /2 min{,2 2}/6. The next emak consies the case whee a numbe of small istances is guaantee by some exta infomation. Moe explicitly, suppose that a paticula polynomial P is given, an afte some computations seveal pais of vey close oots v α, v β,..., v αl, v βl ae obtaine, whee l #E an fo i l, v αi v βi + i with l 0 note that these pais coul possible have common vetices. The iea is to use this infomation to impove the geneal boun fom Theoem 6 by a facto of #E+ + + l. In this way, the close the oots that have been iscovee ae, the moe the boun is impove. It coul be paticulaly useful to boun the minimal istance between iffeent oots when at least two pais of vey close oots have been iscovee, taking E as the set with only one ege joining a pai of closest oots. Remak Following the notation in Theoem 6, suppose that > 2 an that thee exist at least l istinct pais of oots v α, v β,..., v αl, v βl whose istance is less than not necessaily these pais of oots shoul be connecte by eges in E. Fo i l, let i such that v αi v βi 9 + i

10 an enumbe these pais such that Then, if #E < l, l 0. v i v j sdisc P /2 MP #E+ #E+ + + l /2 min{,2 2}/6. Inee, suppose that 0 < ω ω 2 ae the oee istances between pais of oots of P. By the assumptions, fo i l thee ae at least i + i + i. istances less than o equal to an then we have that ωi Consie Ẽ the set of l eges whose lengths ae ω,..., ω l, this is to say, the set fome by the l eges with smallest lengths. Then, applying the boun in Theoem 6 to G {v,..., v }, Ẽ we obtain the following boun fo the pouct of the lengths of the eges in E: v i v j v i,v j Ẽ #E ω i i l v i v j i#e+ ωi sdisc P /2 MP l /2 min{,2 2}/6 l sdisc P /2 MP #E+ #E+ + + l /2 + i i#e+ min{,2 2}/6. Finally, as an application of Theoem 6, we give a simplifie poof of [8, Theoem 9] with smalle constants. Theoem 4 Let P C[X] be a polynomial of egee with exactly 2 istinct complex oots an let V {v,..., v } C be the set of oots. Fo any oot v of P, we enote by sepp, v the istance fom v to one of its closest iffeent oot of P. Then, fo any V V, v V sepp, v sdisc P MP 2 #V min{,2 2}/. Poof: Fo each v V, we take ṽ as one of its closest iffeent oot of P. We consie the multigaph G V, E whee E is the multiset of eges of type v, ṽ with v V. Note that each ege in E can occu at most 2 times one fo each of its vetex. We ivie E in two sets E 0 an E, with E 0 having all the elements in E an E having the elements that occu twice in E. Applying Theoem 6 to V, E 0 an 0

11 V, E an taking into account that #E 0 + #E #V, we obtain sepp, v v V v i v j v i v j 0 sdisc P MP 2 #V min{,2 2}/ as we wante to pove. Acknowlegments: The authos ae helpful to the efeees fo thei vey useful comments an suggestions. Refeences [] S. Basu, R. Pollack an M.-F. Roy, Algoithms in eal algebaic geomety. Secon eition. Algoithms an Computation in Mathematics, 0. Spinge-Velag, Belin, [2] J. Davenpot, Cylinical algebaic ecomposition. Technical Repot 88-0, Univesity of Bath, Englan, 988. [] Z. Du, V. Shama an C. Yap, Amotize boun fo oot isolation via Stum sequences. Symbolicnumeic computation, 29, Tens Math., Bikhäuse, Basel, [4] A. Eigenwillig, Real Root Isolation fo Exact an Appoximate Polynomials Using Descates Rule of Signs, Doctoal issetation, Univesität es Saalanes, 2008 [5] A. Eigenwillig, V. Shama an C. Yap, Almost tight ecusion tee bouns fo the Descates metho. ISSAC 2006, 7 78, ACM, New Yok, [6] I. Emiis, B. Mouain, E. Tsigaias, The DMM boun: multivaiate aggegate sepaation bouns. ISSAC 200, , ACM, New Yok, 200. [7] J. Johnson, Algoithms fo polynomial eal oot isolation. Quantifie elimination an cylinical algebaic ecomposition Linz, 99, , Texts Monog. Symbol. Comput., Spinge, Vienna, 998. [8] M. Kebe an M. Sagaloff, A wost-case boun fo topology computation of algebaic cuves. J. Symbolic Comput , no., [9] D. Kincai an W. Cheney, Numeical analysis. Mathematics of scientific computing. Secon eition. Books/Cole Publishing Co., Pacific Gove, CA, 996. [0] K. Mahle, An inequality fo the isciminant of a polynomial. Michigan Math. J [] M. Mignotte an D. Ştefănescu, Polynomials. An algoithmic appoach. Spinge Seies in Discete Mathematics an Theoetical Compute Science. Spinge-Velag Singapoe, Singapoe; Cente fo Discete Mathematics & Theoetical Compute Science, Aucklan, 999. [2] C. Yap, Funamental poblems of algoithmic algeba. Oxfo Univesity Pess, New Yok, 2000.