The Polynomial Roots Repartition and Minimum Roots Separation

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1 WSEAS TRANSACTIONS o MATHEMATICS Cl Alexe Mues The Polyoml Roots Retto d Mmum Roots Seto CĂLIN ALEXE MUREŞAN Detmet of Mthemtcs Fculty of Lettes d Scece Uvesty Petoleum d Gs of Ploest B-dul Bucuest No39 Ploest Phov ROMANIA cmues@ug-loesto cmues@yhoocom Abstct: It s ow tht f ll the oots of olyoml e el they c be loclsed usg set of tevls whch cot the thmetc vege of the oots The m of ths e s to eset ogl method fo gvg othe dstbutos of the oots/ modules of the oots o el xs method fo evlutg d movg the olyoml mmum oot seto esults method fo the comlex olyomls d fo olyomls hvg ll oots el We use the dscmt Hdmd s equlty Mhle s mesue d ew ogl equltes Also we wll me some cosdetos bout the cost fo solte the olyoml el oots Ou method s bsed o the successve slttg fo the tevl whch cot ll oots Key-Wods: Roots etto Isoltg the oots Mhle s mesue Itoducto The oots etto of comlex olyoml o el xs mes to gve el tevls ot ecessly dsuctve fo evey olyoml oot o module of the olyoml oot Pe-soltg esectvely soltg the comlex oots of olyoml wth comlex coeffcets mes to comute setg boxes the comlex le whch cots t most oe esectvely exct oe comlex oot of the olyoml The exct lgothms fo soltg the oots e bsed o: Stum s sequeces see []; dffeetto techque see [] d Vcet s theoem see [3] Othe oot fdg methods e the umecl methods These methods wo wth get vety of oxmto eos Recetly esults whch geelze the uvte Hemte teolto fomul c be foud [4] [5] [6] The umecl lgothms comute oxmto fo ll the comlex oots of olyoml u to desed ccucy d f tht s smlle th mmum oots seto the the lgothms c be tued o the solto lgothms Fo the sme oxmto eo the costs of the fstest soltg lgothms d umecl lgothms e comble The m of ths tcle s to gve ogl esults egdg the oots etto o el xs the mmum oots seto fo comlex olyomls d fo the olyomls wth ll el oots These esults e ecessly exct lgothms fo soltg the oots bsed o successve slttg (see [] d [] d c be useful lso umecl lgothms I ths secto we toduce the bss ottos d otos d eset the elmy esults ths feld We deote comlex olyoml wth P( x = x + x + + x+ x x d ts oots wth Defto Let be P( x C[ x] We defe: = ( x x b se( P = m{ x x / x x } the mmum oots setos c P = the olyoml om d L( P = the olyoml legth x e + x + + x x= the vege of the oots Defto The dscmt of olyoml P C[ x] wth ledg coeffcet d oots x x s defed s: ISSN: Issue 8 Volume 7 August 8

2 WSEAS TRANSACTIONS o MATHEMATICS Cl Alexe Mues - dsc(p= ( x x Π < Poosto Let be P C[ x] The exesso of D= Π ( ( / D= ( b dsc(p= - D Defto 3 Let be m < x x s x x K x x x K x M M M x x K x m m P= P( x = x + x + + x+ d m Q= Q( x = b x + b x + + b x+ b m m b Sylveste's mtx of P d Q s the mtx S: m m K K m m K K M O O O K O O M M O O O K O M K m m K S= b b K b K b b K b K M O O O K O O M M O O O K O M b b b K K whee thee e ows of the followed by m ows of the S= s s mtx wth m + ows d m + colums hvg the elemets: s = fo d s = b b m fo m Defto 4 The esultt of P d Q s the detemt of Sylveste's mtx es (P; Q=det(S Poosto dsc( P = es( P; P See [7] o [8] Obsevto The dscmt dsc(p c be exessed oly by the degee d the olyoml coeffcets See Defto 3 d 4 d Poosto Fo my othes esults ths e see [9] Theoem ([7] Let be P( x = + x+ x + + x + x ; C; ; wth the oots x x x C; f = mx{ x x x } the: m x = b + mx { } c If λ λ ; + such tht + + λ = the λ / { ( λ } mx mx { } { } / d mx mx e Fo ; { } we hve: mx{ } f / / mx {( g ( } C Coolly Let be P( x = x + x + + x+ P R[ x] the R el such tht x R Poof: Obvously esults fom the Theoem elcg wth fo { } Fo R {} we c te: L( P R = d > = L( P < Fo othes useful bouds see [] d [] Defto 5 The Mhle Mesue of the olyoml P deoted by M(P s: = M[ P( x] = mx{ x } Theoem = Let P( x = x + x + + x+ P( x C[ x] the: π θ l[ ] = l P( e dθ π ISSN: Issue 8 Volume 7 August 8

3 WSEAS TRANSACTIONS o MATHEMATICS Cl Alexe Mues See [] [3] Poosto 3 ([7] Wth the otto fom Theoem we hve = = m{ x } b M[ x P] = M[ P( x] x c M ( P Q = M ( Q ( P Q C[ x] d M[ P( x ] = M[ P( x] e + P P Theoem 3 ([4] Let P( x = x + x + + x+ d P( x C[ x] o P( x = ( x x = wth the oots x x x C; ot ecessly dstct Itoducg the olyomls m m Pm x =± ( x x ; m = we c clculte Pm Method tht s: P ( x = P( x If * m N we c obt m ccodg to Geffe's G x H x P x C x fom the { m m m+ } [ ] eltos: Pm ( x = Gm ( x x Hm( x Pm+ ( x = Gm ( x x Hm ( x Sttg to the evous ots we fd fom ecuso method: Pm ( x fo m Theoem 4 ([4] If P( x C[ x]; deg( P d Pm m the olyoml sees ssocted to the Geffe s Method the: m m m Pm Pm m d lm Pm = Theoem 5 = f{ P Q / Q C[ x] s moc olyoml} Q Fo ovg see [5] d [6] Obsevto The Mhle mesue M(P c be oxmted usg oly the degee d the olyoml coeffcets See Defto 5 Theoem 3 d 4 Defto 6 A e-soltg/soltg tevl fo comlex olyoml eeset oe tevl ( b hvg s lmts two tol umbes betwee whch thee s t most/ecsely oe oot (modules of the oot of the olyoml Fo soltg tevl we hve: ( N such tht { x x x} ( b = { x} fo el oots of the olyoml d N such tht x x x b = x { } { } fo comlex oots of the olyoml b Pe-soltg/soltg the el oots cossts fdg fo ll the olyoml s oots dsuctve e-soltg/soltg tevls Defto 7 Fo gve fucto g(x g : R R we deote by O(g(x the set of fuctos: O(g (x ={f(x / f : R R ( c x R such tht f(x c g( x ( x x } I ths cse fo evey f(x we deote: O(g(x=f(x We e syg tht f gows t the sme te o t my gow moe slowly th g whe x s vey lge Defto 8 Fo gve fucto g(x g : R R we deote by (g(x the set of fuctos: Θ(g(x= { f(x / f : R R O(g(x=f(x d O(f(x=g(x} I ths cse fo evey f(x we deote: Θ(g(x=f(x We e syg tht f gows t the sme te th g whe x s vey lge Fo moe detls see [7] d [8] Polyoml oots dstbuto Ceteed tevls etto method fo olyomls wth ll el oots Defto Let be b b el umbes We deote by P ( b b R [ x ] the set of ll olyomls hvg the fom: P( x = x + b x + b x + + b N wth ll el oots d the followg equltes hold betwee the oots: x x x x Poosto If P( x P ( b b the: = x ( x ; = = b b = ( b b ; c x= See [9] d [] Theoem Let be P ( x = x - x+ x x ( { } the P ( x P ( b b ISSN: Issue 8 Volume 7 August 8

4 WSEAS TRANSACTIONS o MATHEMATICS Cl Alexe Mues See [9] d [] Theoem Let be P( x P ( b b the: x+ x x+ ( b x ( x x ( c x x x+ + ( 3 { } If we deoted by {( y ( y ( y3 ( y } the oots of P ( x fo fxed { -} to the Theoem the the evous delmttos e otml the set P ( b b so the exst tul umbe such tht: m{ x } = ( y m{ x } = ( y ; P P ( b b P P ( b b mx{ x } = ( y mx{ x } = ( y ; P P ( b b P P ( b b See [9] d [] A dffeet dstbuto fo the olyomls P( x P ( b b oots Theoem Let be P( x P ( b b Be t: α= ; = ( ; α = ( ; = ; α = ; = + = I = [ α ] fo = The we hve: Fo = x I = [ x+ α x + ] b Fo {3 }: + = α + I I [ ] c Fo {3 }: α = + d α = α= d α < > α < α e x [ x+ α x+ ] {3 } { x α } { x } { x } { x } I I = { φ} I I = + α = + - I I = + = + I I esectvely I - we c foud ll the olyoml oots but x esectvely x c be oly t the lmts of the tevls f I the tevl [ x+ x+ ] esectvely the tevl [ x+ α x+ α( ] we c foud t most the oots { x x x x } esectvely { x x x x } tht s t most oots fo + + { } 3 Poof: x I = [ x+ α x+ ] = s obvous fom evous theoem b Fo {3 }: > + + > / ( + > + α > α + > + < / ( + ( > + The both eltos e tue c α = = obvous α = = obvous + α = + = lso s obvous fom the theoem ottos α < d ( ( > ( > > Also we c ove tht: α = ( < α = e Fom the ot b d d: α > α+ > + fo {3 } Now the fst elto s mmedtely The othes esult fom ot c f We c obseve tht x x d oly these oots c be [ x+ x+ ] The x x d x 3 d oly these oots c be [ x+ 3 x+ ] d fom the sme oceed we obseve tht { x x x x } d oly these oots c be [ x+ x+ ] Smlly { x x x + x ( } d oly these oots c be the tevl [ x+ α x + α( ] ISSN: Issue 8 Volume 7 August 8

5 WSEAS TRANSACTIONS o MATHEMATICS Cl Alexe Mues Theoem If P( x P ( b b fo x x x x the oots of P we hve: 3 se( P the equlty s elsed fo + 3 P( x = x x = b x x + See [9] d [] fom Refeeces Theoem 3 ( {3 } the legth of the tevl [ x x ] + + o [ x+ α x+ α( ] s = d < + b Fo 3 s ossble tht se( P c A suffcetly codto fo hvg t most oe oot the tevl [ x+ x+ ] s: + se( P Poof: + = ; = ( {3 } the: + = -+ ; ; - ; - fo 3 Hece { } + < ( + ( ( = So < Fom the evous theoem ot d [ x x ] + + esectvely + α + α e the sme legth [ x x ( ] b If se( P the fom Theoem 3 we hve se( P Fom the evous ot we hve : < Suosg 3 we c obt 3 c If se( P the the tevl [ x x ] + + we c hve t most oe of the olyoml oots The fom evous ot the esult s mmedtely Theoem 4 Let be P( x P ( b b the we c toduce the olyoml Q( x = P( x+ x d H ( x = x Q( x > tul d we hve: H( x P + c c ( whee c c R b If ( y {3 + } e the oots of H(x such tht: y+ y+ y+ y y the: y= x s the vege of the oots of H The b = ( + b ( + b dscmt of H s =+( b o c If we deote: α = ; = ( + ; α = ( + ; = ; α = = ; = the + the oots of the olyoml H: y [ α ] + d I the tevl [ x+ x+ ] esectvely tevl [ x+ α x+ α( ] we foud t most the oots { x x x x } esectvely { x x x x } so oots fo ( {3 +} Poof: b Obvous c d See Theoem e fo x P( x P+ ( c c Theoem 5 Wth evous ottos fo {3 +-}: < d < > d < ISSN: Issue 8 Volume 7 August 8

6 WSEAS TRANSACTIONS o MATHEMATICS Cl Alexe Mues > d > fo {3 } v Fo < < tul > Fo > f exst {3 + } fxed such tht ( x [ ] the: { x } [ ] [ ] v Fo {3 + } < + suosg fxed d ot deedg t the: lm = b b Poof: < + < Sml evous theoem obvous < We obseve the logy wth + + > > + + > + b b But = ( > d s eough to ove tht: ( + ( + ( + > + ( + Fo smlcty we deote +=x d we obt: x ( x ( + > x x ( x x ( > x x ( + 3 x x ( x + x( + x 3 ( > x ( x ( 3 x + x ( + x( + > 3 But x x ( + = x [ x ( + ] > becuse x= + > + {3 } d x( > ( x > fo x> < s mmedtely fom < elcg wth - > + > ( + ( + ( + > + ( + If we deote +=x we obt: x x ( + > x x The ext elto s esy to ove whe we obseve the ostvty of the deomtos ( b ( b ( b ( b + + = = + + ( x b ( x b x = < ( x b xb x The suffcetly elto fo ove become: x x ( + x > x ( x x x x ( + > > x x Now > s obvous fom evous elto elcg wth - v Fom > > d fom fo < < The ( ( { φ} - - < The fom hyothess we hve x [ ] Suosg x [ ] we hve cotdcto wth: x [ α ] d the suosto s flse v Results fom Theoem 4 b c: =+( b b = ( + b ( + b α = = ; = the + + < = + + ISSN: Issue 8 Volume 7 August 8

7 WSEAS TRANSACTIONS o MATHEMATICS Cl Alexe Mues Suosg fxed we obt: lm + = lm + ( + b ( + b = b b b lm = b Obsevto Fom the lst theoem we obseve the dstbuto fo umbes c see fg: s we - fg Obsevto If we fd oe of the tevls of Theoem 4 d sgle oot of the olyoml we c edefe the tevl usg the elto { x } [ ] [ ] ( > ( > tuls d tg lge ; see Theoem 5 v Alcto Fo olyoml wth ll el oots d wth degee =3 x I = [ x+ α x+ ] = 3 d the tevls e soltg tevls The oof s esy to me sttg to the Theoem I the geel cse fo el olyoml f P α P > the { 3} such tht olyoml wll hve oly oe el oot d P( α P( < { 3} so tht Theefoe we deteme the tevl whch cots the oot x I I both cses we c edefe the tevls cotg oots smly by dvdg them d usg cotuous oety fuctos o usg lst Theoem 4 v d toducg the olyomls Q( x = P( x+ x d H ( x = x Q( x whee > tul My ctce ocesses use o c be modelled wth the hel of the el oots of the olyomls wth smll degee see fo exmle elto (7 [] 3 A oots dstbuto fo comlex olyomls Theoem 3 Fo bty comlex olyoml P( x = x + x + + x+ wth d fo { } such tht x x x x x the + / / P x fo d + + x P fo + Poof: Fom Defto 5 d Poosto 3: mx{ x } = P = P ( m { x } The x x x = ( Becuse x fo the x Suosg x > the x x > But x x d the x x x x x > cotdcto wth ( The the suosto s flse d Usg the ducto method the x x (3 Sttg fo (: x+ x = Becuse x fo + the x Suosg ISSN: Issue 8 Volume 7 August 8

8 WSEAS TRANSACTIONS o MATHEMATICS Cl Alexe Mues x < the x x+ < d the x x < d x+ x x x x < cotdcto wth ( The suosto s flse d By ducto we c ove tht x x + fo o + x fo + (4 Now becuse P we hve the esult Coolly 3 If > x x the = fo : + + / x < = P Poof: = mx{ x } = = We c eet the ocedue bove d the + + = x < fo Poosto 3 Fo l + < < l + γ + whee γ γ 577 s Eule costt (5 Poof: It s well ow see fo tht d fo othe sml equltes [] tht lm l + c = γ whee ( c el = lm c = γ 577 s Eule costt (6 u = l γ v = l γ e obted fom the elto bove (7 + The u+ u = + l ( + + v+ v = + l (8 + + x+ x Fo the fuctos u( x = + l ( x+ x x+ x v( x = + l x> x+ x+ u '( x = < v'( x = > ( x+ x x ( x+ Becuse lm u( x = lm v( x = Fo > x x the u = u+ u > v = v+ v < ( u s stctly cesg u < γ ( v s stctly decesg v (9 > Fom (7 d (9 we hve the esult Obsevto 3 I the evous esults we geelse the equlty: ( / / x P γ fo see [3] fom efeeces Theoem 3 Fo bty comlex olyoml wth degee d the ledg coeffcet fo { } so tht x x x x+ x the: l l + γ + ( l l + l whee γ 577 s Eule costt Poof: Fom theoem 3 we hve: + + = x x x = + ( But l + γ < < l + γ + whee γ 577 s Eule costt ( ISSN: Issue 8 Volume 7 August 8

9 WSEAS TRANSACTIONS o MATHEMATICS Cl Alexe Mues = < l( + γ ( = <l + γ + l + ( >l + (3 + The fom ( ( d (3 we hve the esult Obsevto 3 The evous theoem c be useful whe we ty to evlute N the umbe of the oots wth modules bgge o equl wth oe Aothe sml theoem s the ext oe Theoem 33 Fo comlex olyoml P( x = x + x + + x+ wth * ; whee N such tht: x x x x+ x > the L( P l see [4] [5] 3 Mmum Roots Seto Theoem 3 If P( x = + x+ x + + x ; P C[ x] s sque fee the: + / se( P > dsc P P + d fo P Z[ x] : se( P > P / 3 / b Fo P Z[ x] : se( P > e P (l + c se( P > m( dsc( P l( 3 { + L P } + se( P > { [ L( P + ] } To ove see [7] fo b see [6] fo c see [3] fom efeeces Fo othes equltes ths e see [4] Poosto 3 Let be f g h : + + f ( x = + x+ x + + x g( x = + x+ 3 x + ( x γ f ( x h( x = The f g e mootoclly g( x cesg fuctos g x x f ( x x x f ( x = x x= = ' ' d + + x ( x + 3 ( x x x g( x = + fo x 3 ( x ( ( fo x= 6 The oof c be doe fom clculto Coolly 3 Wth the evous ottos: h( x h( = 6 ( ( ( Poof: We c obseve tht: f g h : x+ x + x h( x = (4 + x + 3 x + ( x + x+ x + x h( x ( + ( x+ + ( x f ( 6 h( x h( = = ( g( ( Theoem 3 Fo P( x = + x+ + x ; R el umbes such tht x R d fo < x x el fo ( c R the [ ] such tht: l= ( = = dsc( P h( c se( P + + l (5 Poof: Suosg x x x d usg the fuctos f g h the Hdmd s equlty d Poosto we c obt: ISSN: Issue 8 Volume 7 August 8

10 WSEAS TRANSACTIONS o MATHEMATICS Cl Alexe Mues ( / D= ( K x x x x x K x M M M x x K x D S T whee S= x x d = T = x (6 = = S= x x ( + x + x x + x x + x x + x Fo < x x d x + x x d usg the tgle equlty we c wte: f ( x ( x S x x g x T = = (7 f Fom (6 d (7: ( ( x h x D x x (8 f h( x s cotuous fucto o exst c [ R ] so tht: ( = ( R ] d m h x / = h c (9 The becuse x { } = ( = = ( x ( x f = ( x ( Usg the Vete s eltos we c obseve tht ( x = ( x ( x + ( x = ( x ( x + ( = = ( x ( x = ( x x x x + = + x + + = < < { } ( + x + x + x + ( = ( ( x x x + x + = < { } ( x + + l whee l= (3 Fom ( ( (3: + + l f ( x ( (4 = = Fom (8 (9 (4: ( = = D h( c + + l x x whee < (5 - dsc P = D (see Poosto Usg (5 we obt the esult Theoem 33 Fo P x = x + x + + x + C[ x] ISSN: Issue 8 Volume 7 August 8

11 WSEAS TRANSACTIONS o MATHEMATICS Cl Alexe Mues f exst { } such tht x x x x x the: + l ( dsc( P γ+ + se( P P b If P( x P ( b b the ( dsc( P se( P γ whee γ = mx{ + α + } Poof : Fom the evous theoem elto (: f ( x D x x f ( x (6 g( x D f ( x The x x (7 g( x f ( x f ( x h( c h( c = m{ h( x / } (8 g( x Fom Coolly 3 h( x h( c ( dsc P The - (9 = D (3 f ( x = f ( x f ( x + + f ( x x f ( f x x (3 Now usg Theoem 3 we obt: f ( x ( f ( x ( f ( x γ+ l + ( f ( x (3 The fom (7 (9 (3 d (3 we obt the esult b f x x + f ( x x γ fo γ α = mx + + (33 see Theoem Relcg (3 wth (33 the demostto to the fst ot we hve the esult Theoem 34 Fo P x = x + x + + x+ C[ x] If x x x x x x > the dsc( P se( P ( Poof: We follow the stes to the evous theoem elcg (3 wth the elto f ( x (34 Obsevto 3 I the evous theoems we c use the elto: dsc( P fo P Z[ x] we obt sml esults fo teges olyomls 4 Isoltg the oots Coclusos Rem 4 If we come the esults fom the sectos wth ou esult fom oe of the dvtge of the secod och s we c see Theoem f s tht we c edct the oots d the mxmum umbe of the oots whch c be the tevls [ x+ x+ ] esectvely the tevls [ x+ α x+ α( ] d we c gve the legths of the tevls Also we c edefe the tevls cotg oots See Theoem 5 v v cetg othe olyoml d owg tht fo Q( x = P( x+ x d ISSN: Issue 8 Volume 7 August 8

12 WSEAS TRANSACTIONS o MATHEMATICS Cl Alexe Mues H ( x = x Q( x > tul exst > tul so [ ] The we hve: tht x > { x } [ ] [ ] Ou esults Theoem 3 usg Mhle s Mesue eeset tul equltes fo boudg evey module s oot of the olyoml d gvg the oots etto fo comlex olyomls Usg these tul wy we obt ew theoem wth the best equltes fom the method eseted Theoem 3 bout the umbes of the oots tht e outsde of the ut ccle We c come the theoem wth oe of Szego s theoem Rem 4 Fo comg ou esult fom Theoem 3 of the mmum oots seto fo the olyoml hvg the oots < x fo wth the othes we c te s we c see Coolly 3 h( c (35 The ( = = C C = + + l L( P L( P l= C> tcully coveet (36 Fom the theoem we c ove fo tul get umbe: C dsc( P se P (37 L P 3/ Ou esult cots whle ll the othes s es e whee s(x s el cotuous fucto But the olyoml hve ll < x R oots el d ostve d To obt esult whee the oots e ot ostve we c ly the theoem fo the olyoml Q( x = P( R x R> x whee se (Q=se(P b Now we c obseve fom the evous elto Fo the tege olyomls we hve dsc( P tg d R s we c see Coolly the R O log = O( l L( P (38 se( P s the ode fo the umbe of successve slttg of the tevl [ R ] [ R] utl we ccomlsh the oot e-solto We c ecse the evluto ow the geel cse fo the successve umbe of slttg: R O log = O( l [ L( P ] (39 se( P fo moe detls see [8] [7] [8] [9] c The cost fo soltg the oots the cse of the comlex olyomls whch s the umbe of the thmetc oetos eeded s domted to the umbe of successve dvsos multled by the cost of Stum s sees ssessmet see [] [8] [7] [9] [3] o by the cost of olyoml evluto ot see [8] o by othes umbes of oetos see [] Fo the olyomls wth ll oots el we me the dvsos of the tevls: [ x+ x+ ] d [ x+ α x+ α( ] to the evous secto d we obt the mmum oetos of slttg the we ly Stum s Theoem o othe methods fo soltg the oots Rem 43 I Theoem 33 Theoem 34 we gve ew esults bout mmum oots seto fo comlex olyomls d fo olyomls wth ll el oots Oe of de s to use the bouds fo modules of the oots gve Theoem 3 Refeeces: [] Hedel LE Itege Athmetc Algothms fo Polyoml Rel Zeo Detemto Jou ACM Vol 8 No 4 Oct [] Colls GE d Loos R Polyoml el oot solto by dffeetto Poceedgs of the thd ACM symosum o Symbolc d lgebc comutto SYMSAC [3] Ats A A mlemetto of Vcet's theoem Numesche Mthemt [4] Boo C d Ro A O the eo multvte olyoml teolto Mth Z 99-3 [5] Sm D O some Hemte Bvte Iteolto Schemes Wses Tsctos o Mthemtcs Issue Vol 5 Dec [6] Sm D A Geelzto of the Remde Multvte Polyoml Iteolto Wses Tsctos o Mthemtcs Issue Vol 3 J 4-6 [7] Mgotte M Itoducto to Comuttol Algeb d Le Pogmmg Ed Uv Bucuest 77-8; 3-35; 49 [8] Dveot J H Comute Algeb fo Cyldcl Algebc Decomosto Techcl Reot 88 School of ISSN: Issue 8 Volume 7 August 8

13 WSEAS TRANSACTIONS o MATHEMATICS Cl Alexe Mues Mthemtcl Sceces Uvesty of Bth Egld 3-5 [9] Bost A Flolet P Slvy B d Schost E Fst comutto wth two lgebc umbes Resech Reot 4579 Isttut Ntol de Recheche e Ifomtque et e Automtque Oct g [] Stefescu D New bouds fo the ostve oots of olyomls J Uv ComSc [] Stefescu D Iequltes o Ue Bouds fo Rel Polyoml Root I: Gzh VG My EW Voozhtsov EV (eds CASC 6 LNCS vol 494 Sge Hedelbeg [] Psolov V V Polyomls Moscow Cete fo Cotuous Mth Educto [3] Mueş CA The Aoxmto of Polyoml s Mesue wth Alctos towds Jese s Theoem Geel Mthemtcs Vol 5 o4 Ed by Detmet of Mthemtcs of the Uvesty "Luc Blg" Sbu Rom [4] Celeco L Mgotte M Ps F Comutg the mesue of olyoml Joul of Symbolc Comutto Vol4 Issue Acdemc Pess Ic Duluth MN USA [5] Mgotte M Some Useful Bouds Comutg Sulemetum 4 (ed B Buchege Colls GE &RGK Loos Sge-Velg We-New Yo [6] Dégot J Fte-Dmesol Mhle Mesue of Polyoml d Szegö's Theoem Joul of Numbe Theoy vol 6 Issue Febuy 997 Elseve 4-47 [7] Kuth Dold E The At of Comute ogmmg vol I Fudmetl Algothms secod edto - Ed: Addso Wesley 98 [8] Come Thoms H Leseso Chles E Rvest Rold L Itoducto To Algothms The MIT Pess McGw-Hll Boo [9] Lus A Metode Numece Ed Costt-Sbu [] Lguee EN Oeuves- I Ps -898 [] Assmbo C Bb A Dozo R d Bue M J Detemto of the Pmetes of the S-Electode Imedce Model fo ECG Mesuemet Poceedgs of the 6th WSEAS It Cof o Electocs Hdwe Weless d Otcl Commuctos Cofu Isld Geece Febuy [] Rum SM Polyoml Mmum Root Seto Mthemtcs Of Comuttos Vol33 N45 J [3] Veescu A O the Sml Ode of Covegece of two Adcet Sequeces Relted to ζ(/ Wses Tsctos o Mthemtcs Issue Vol 5 Decembe [4] Szego G Othogol Polyomls Ame Mth SocColloq Publ Volume XXIII New Yo959 [5] Mueş CA Szego s Theoem Sttg Fom Jese s Theoem UPB Scece Bull Sees A Vol 7 No 4 Ed by Uvesty Poltehc of Buchest [6] Colls G E Hoowtz E The Mmum Root Seto of Polyoml Jsto Mthemtcs of Comuttos Vol 8 No [7] Mueş CA Bucu A The cost of the tol olyoml's oots'seto Studes d scetfc esech mthemtcs sees Vol 7 sulemetum Ed by Detmet of Mthemtcs of the Uvesty Bcǎu Rom [8] Mueş CA The cotuous fuctos techques fo soltg the oots of tege olyomls Studes d scetfc esech mth sees Vol 8 Ed by Detmet of Mthemtcs of the Uvesty Bcǎu Rom [9] Ems Ios Z Tsgds Els P A ote o the comlexty of uvte oot solto Isttut tol de echeche e fomtque et e utomtque No 643 Novembe 6 g 5 [3] Du Z Shm V d Y CK Amotzed boud fo oot solto v Stum sequeces It Wosho o Smbolc Numec Comutg edtos I D Wg d L Zh School of Scece Behg Uvesty Beg Ch ISSN: Issue 8 Volume 7 August 8

Studying the Problems of Multiple Integrals with Maple Chii-Huei Yu

Studying the Problems of Multiple Integrals with Maple Chii-Huei Yu Itetol Joul of Resech (IJR) e-issn: 2348-6848, - ISSN: 2348-795X Volume 3, Issue 5, Mch 26 Avlble t htt://tetoljoulofesechog Studyg the Poblems of Multle Itegls wth Mle Ch-Hue Yu Detmet of Ifomto Techology,