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Real Algebraic Numbers: Complexity Analysis and Experimentations

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(1)Real Algebraic Numbers: Complexity Analysis and Experimentations Ioannis Z. Emiris, Bernard Mourrain, Elias P. Tsigaridas. To cite this version: Ioannis Z. Emiris, Bernard Mourrain, Elias P. Tsigaridas. Real Algebraic Numbers: Complexity Analysis and Experimentations. Reliable Implementations of Real Number Algorithms: Theory and Practice, 2008, Dagsthul, Germany. pp.57-82. �inria-00071370�. HAL Id: inria-00071370 https://hal.inria.fr/inria-00071370 Submitted on 23 May 2006. HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés..

(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. Real Algebraic Numbers: Complexity Analysis and Experimentations I.Z. Emiris — B. Mourrain — E. Tsigaridas. N° 5897 Avril 2006. N 0249-6399. ISRN INRIA/RR--5897--FR+ENG. Thème SYM. apport de recherche.

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„|‡‹Œ rj_5¯/Œ5u¥_Gdm q]u¦m Œ kpu¥r‰°G‡_©kŠmo„;„|mªd^`m)k r‰ckŠm , cτ 0¶ ¶ /Ukpu¥]2š 13154ª§ M (τ ) = O (τ lg τ ) „t‡‹ M (d, τ ) = O (dτ lg (dτ )) ¬¹rtz"kpx]u¥mo„tq]s¥_ 6 Ÿ * Ð Ÿ Ï"Ð 'Ÿ 7¾ -98@„tzpuvrtx‡kL„|svštrtzou¥mp[]^/k@_5¨luvkŠmL¬¹r‰zL†+rtsvid‡rt^`uv„ts‡zo_G„ts‡zordrtmLu=kŠr‰sv„|mpuvrt¾§dq]xlm&^ªrdknm&r|¬Émo[]_5^ ¬¹rlŒ x‡k rtkow‰xˆ„|zo_ ±F¬¹zp_G_~†+rtsvij]rt^`u=„|s=k5¶ "rtsvsvu¥‡k>„|ˆ‹µ ¼jzou¦m)„t: k $à+“ &Tuvdmpzorj‹]x‡Œ _P‹µ„'zo_G„tszordrtm~u=kŠr‰sv„|mpuvrtµ„ts¥š‰rtzou¦mo[]^ q‡„tkp_G‹r‰‚~_GkoŒ5„|zpmp_Pk5҈zpx‡s¥_"r|¬ kuvpu¥š‰ª[]uv­ku¥†]mp[©zprjŒ rtrt¬&^`­ †]„tsvkB_ ¨lŒ5u¥rtmnizozpO_PeŒ.mp_P(d‹«qjτ?i )>7¶TzoZ „t[]‡_U‹lu=qˆŒ)r‰@¼ x]‡$ —‰‹`+— &³­ ¶©„tƒkTr‰uv^`x]u¥†]svsvzpu¥_Gr;zb–‰_G„|‹"‡qj@‹ <i A+;‰uvrt^`[]^`ˆkŠ_5r‰zo^`= „t$ؗ|]˜ ý&‡mp¿Ár Œ B¬ Oe$ ‰' (d&„|τ‡‹») zo„|_ ‡¬¹‹`_Gzp„B_G‡šdŒ „|_P† k mp[‡_5zo_5uvˆÀ~†]zp_PkŠ_Gdmp_G‹„ºx‡]u ˆ_G‹»„|†]†]zor‰„‰Œ)[µ­u¥mp[»rt†lmou¥^/„|sT^`_G^ªr‰zpi^/„|ˆ„|št_G^`_5dm~¬¹r‰z7–Ö„tzpuvrtxˆk>„ts¥š‰rtzou¦mo[]^/k mp[ˆ„;mU‹l_G†ˆ_G‡‹rt‚U_PkpŒG„|zpmp_GkGÒlzox]s¥_7rt¬kpu¥š‰¾¶  4„|svštr‰zpu¥mp[‡^6¿Á­_”ŒG„|svs&u¥Dm CFEGFHIJEKLNM'"¬¹zpr‰^ ]r;­ rt+Àmo[‡„;m/u=k`q‡„tkp_G‹Drt „Œ5rt^bq‡u¥‡„|mpuvrt r|¬ ‚~_GkoŒ5„|zpmp_Pk5҇zpx]sv_b„t‡‹”rtmo[]_†]zort†+_5zpmpuv_GkUr|¬@´&_5zo‡kŠmp_5uv”q‡„‰kŠu=k ‡z)knm>„|†]†+_G„tzp_P‹u¥0 $Ø— OP&8„|ˆ‹u¦m)k~Œ5rt^`†]sv_ ¨lu¦mni ‡z)knmr‰qlmo„tu¥]_P‹ºuv2  $ RQF&³¶@Z []uvk ^`_5mp[]rl‹kŠ_G_5^/kmpr/[‡„Ö–‰_Umo[]_7qˆ_PknmUŒ5rt^`†]sv_ ¨lu¥mni'u¥†]z)„tŒ mpu=Œ _t¶TZ []_Buvdmp_Gzp_Pknmo_G‹ zo_G„t‹]_5zL^/„Öi'„tsvkprbzo_ ¬¹_GzLmoDr $ؗ|'’ &W¬¹r‰z&„"–;„|zouv„t‰m&­u¦mo[ºr‰†lmpuv^/„|sW^`_5^`rtzoi`^`„t‡„|š‰_5^`_5dmP¶aj_G_>„tsvkpSr $ؗt'— &¾·¸'mp[‡_ ko„|^`_BŒ rtdmo_ ¨jmG§ !@uvšt_Gj­u¥svs¥uvš`_ mU„tTs $UQÖ+— &†]zort†+r‰kp_G‹©„`zo„t‡‹lrt^`uv°5_P‹º„|svštr‰zpu¥mp[‡^ ¬¹rtz kowdx‡„|zo_ ±F¬¹zp_G_U†+rtsvij]rt^`u=„|s=k ­u¥mp[4q]u¥m/kŠmpzo_G„t^ Œ rj_5¯/Œ5u¥_GdmokG¶ Z []_”Œ5rt^`†]sv_ ¨lu¦mniDr|¬>„ts¥s&mo[]_Gkp_µ„|svštr‰zpu¥mp[]^/k"uvk`q+rtx‡‡‹l_G‹ qji Oe (d τ ) ¶ ƒ_PŒ _5dmos¥i‰V§ 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(25)  FR

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(27) LE  LtqRL vuwqRE. k Y. i=1. qRE. p. τ. i. ]N^. τ (1−d). i. 2egf on. 0p. Ih. 1. m. 1. <q k?q. i. d. |αi − βi | ≥ M(f )−d+1 d− 2 (. k. 1. i. √. 3 k ) d.  FH JEH.  L LHLh. f Ÿ оР¢ /Ukp_$ؗ|˜'&¿Ártz$ ORQ|§‡—P'&¹À „|‡‹®¿ QPÀ ¶ £ _„|zo_~š‰rtuv]šbmor'‹l_ m)„|uvsɍ]r;­ mo[]_B‹lu%+_Gzp_G‰mUkŠmp_G†‡k r|¬8mp[‡uvk š‰_5]_Gzo„tsWkoŒ)[]_G^ª_‰§ ‡z)knmx‡kpu¥‡š/ajmpx]zo^ kp_Gwdx]_G‡Œ _P2k „|ˆ‹º]_5¨jm x‡kŠuv]š/´&_5zo‡kŠmp_5uvkŠr‰s¥–‰_5zP¶

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(29)   "( b  8 £ _ªzo_GŒG„|svs0[‡_5zo_mp[‡_ª^/„tu¥u¥]š‰zp_P‹luv_5dmok~zp_Gsv„|mp_P‹”mpr”admox]zp^¤kp_Gwdx]_G‡Œ _`Œ r‰^`†]xlmo„|mpuvrt‡k>„|ˆ‹”mp[‡_5uvz>q‡u¦mŒ r‰^ª± †]sv_ ¨lu¥mnit¶. 6. x. {z:|. y. S€. s. ~}. ‚@ƒ „k†ˆ‡Š‰,‹ŒHŽ‘’Eƒb“”F•S„k”F–k‘M”F“. ¾€ _5m A = P a X , B = P b X ∈ Z[X] ­[‡_5zo_ deg(A) = p ≥ q = deg(B) „t‡‹ L (A) = ¶ £ _ ‹l_5‡r|mp_qji rem (A, B) „|‡‹ quo (A, B)À8mp[]_zo_5^/„tu¥‡‹]_5z@„|ˆ‹bmo[]_ wdx]r|mou¥_GdmG§tzo_Gkp†+_GŒ.mou¥–‰_5svit§ L (B) = τ r|¬8mp[‡_ !Lx‡Œ svuv‹]_G„|”‹luv–juvkpuvrtr|¬ A qdi B §]uv Q[x] ¶ Ì , *¹ ž * Ð ,  R7kŠuvšt]_P‹†ˆr‰s¥ij]r‰^`uv„tszo_5^/„|uv‡‹l_GzUkŠ_Pwdx]_5‡Œ5_=L A E  B SPRS (A, B)   H R p k=0. c. X—. k. ™˜š›œ-. RLonE L< <pRE  j. α1 > β1 = α2 > β2 = . . . αk > βk := αk+1 r. G uwqG uwq. k. q k=0 k. k. Mq. h. MqRw‰x‡r|mpuv_5dmkp_Gwdx]_G‡Œ _Lh E   HqG~wdx]r|mou¥_G‰mq+rjr|m  . r. r. ?q. R0 = A, R1 = B, R2 = − rem (A, B) , . . . , Rk = − rem (Rk−2 , Rk−1 ) rem (Rk−1 , Rk ) = 0 s Qi = quo (Ri , Ri+1 ). E .  .HqG)jRLonE L<%<pRE. A B (Q0 , Q1 , . . . , Qk−1 , Rk ) s. {Qi }0≤i≤k r.

(30) ™. 5<JF  LFJEVF YF P. Z []_Gzp_Tu=k„ [jx]š‰_@q]uvq]s¥uvrtš‰zo„t†][ji rtBkpu¥š‰]_G‹7†+rtsvij]rt^`u=„|s|zo_5^/„tu¥‡‹]_5zkp_Gwdx]_G‡Œ _Pk@¿ÁŒ ¬ $ Qt+§ t’l § ORQF&j„|‡‹>zo_ ¬¹_Gzp_G‡Œ _Pk mp[‡_5zo_&uvˆÀ ¶5$ O‰˜&]†]zp_PkŠ_Gdmp_G‹ª„>x]]u‡_G‹`„t†]†]zor‰„tŒ)[mprBkŠx‡q]zp_PkŠx‡s¦m)„|dmokG¶0‡rtzmp[‡_Uadmpx]zo^ª±³²~„|q]u=Œ)[dm ¿¹rtzLalidsv–t_Pknmo_5zp± ²U„tq]u=Œ)[‰m.À kŠ_Pwdx]_5‡Œ5_Gkmp[‡_Bzp_P„t‹l_Gz&^/„Öi'zo_ ¬¹_Gz mpZr $ QG'‘ &³¶ ·¸½mo[]u=k`†‡„|†+_5zª­_Œ r‰‡kpuv‹l_Gzªmp[]_”admpx‡zp^ª±¸²U„|q‡uvŒ)[dm/kŠ_Pw‰x‡_5‡Œ5_©rt¬ „|‡‹ B §LuF¶ _ StHa(A, B) §L­[‡uvŒ)[ Œ r‰dmo„|uv‡k>†+rtsvid‡rt^`uv„tsvk~mo[‡„;m„|zo_ª†]zpr‰†ˆr‰zŠmou¥r‰‡„|s8mor©mo[]_`†ˆr‰s¥ij]r‰^`uv„tsvk>Auv SPRS ¶'admox]zo^"±¸²U„tq]u=Œ)[‰m kp_Gwdx]_5ˆŒ _GkB„‰Œ)[]uv_5–t_`q+_ mpmp_5z"qˆr‰x]‡‹]krt»mp[]_ºq]u¦mªkŠuv°5_/rt¬mo[]_ºŒ rj_5¯/Œ5u¥_GdmokB„|ˆ‹[ˆ„Ö(A,–t_`B)štrjrl‹®kŠ†+_GŒ5uv„ts¥uv°G„|mpuvrt †]zort†+_5zpmpuv_GkG§]kŠuv‡Œ5_~mo[]_5i„|zo_>‹]_ ‡]_P‹©mp[]zortx]š‰[‹]_ mp_Gzp^`uv‡„|dm)k5¶ €¾_5m M§ q+_mp[‡_b^/„;§ mozpu¥¨©­[]u=Œ)[µ[‡„‰k „tkUzpr;­ kmp[]_ªŒ rj_ ¯'Œ ­uv_5u¥dmp[BmUzo–‰_G_GkpŒ †+mp_GrtŒ.z)mÉkmortr ¬8mpmo[][]_L_b^ª†+r‰rt]svijrt]^`rtu=^`„|s;u=„|q‡s=„‰k kŠAX § § = u k AX . . . , AX, A B, BX, . . . , BX BX X ¶0Z []_&‹lu¥^`_G‡kŠuvrtBrt¬ M u=k ,(p+q−1−2j)×(p+q−1−j) ¶WÂ]r‰z l = 0, . . . , p+q−1−j X , . . . , X, 1 sv_ m M qˆ_7mo[]_kpwdx‡„tzp_>^/„;mozpu¥¨ºr|¬T‹luv^`_5‡kpu¥r‰ (p + q − 2j) × (p + q − 2j) rtq]mo„|uv]_P‹ºqji/m)„|¼ju¥‡š"mo[]_ ‡z)knm Œ rtsvx]^`‡k„t‡‹'mo[]_ l±Fmp[”Œ5rtsvx]^`©rt¬ M ¶ p + q − 1 − 2j Ì , *¹ ž * Ð ,  G H F P

(31)  %H  RE   L A E  B  H G: RE  q−1−j. j. q−2−j. p−2−j. p+q−2−j. p−1−j. p+q−1−j. j. l j. j. c. X—. Mq. <q. <p. h. q. r. <p. StHa(A, B) = (Hp = Hp (A, B), . . . , H0 = Ho (A, B)). G E  H = Pj det (M l )X l s MqR <pRE <LhSj FJE jG H F  P

(32) qH =B j l=0   Y

(33) .Ej  ' L  KHERpH  =(hA, H=p−1 E    H?qR L  ERHLh JE h (A, B), . . . , h B)) h = H1qGE HqGhj<pRE   JU  Y!hXH jJM HqG jGLo nEL p H r phL 0 ≤ j ≤ p 0s(A, qRE hL:LWp hj = 0 r j LH qG  uT  ELE  P  hjH JM' s uwq. ·³¬ StHa(A, B) u=k8]rt]±³‹l_5¬¹_GŒ mpuv–t_@mo[]_5"u¦mTŒ r‰u¥ˆŒ u=‹l_Gk0x]†mor>kpuvšt­u¥mp[bmp[]_ Œ s=„tkokŠu=Œ5„tsdkpx]q]zo_Gkpx]s¥mo„t‰mkŠ_Pwdx]_5‡Œ5_t¶ ² r;­&_5–‰_5zP§u¥Dmp[]_‹l_ ¬¹_PŒ.mou¥–‰_ŒG„tkp_t§Trt‡_©[‡„‰kbq+_ mpmp_Gz`Œ5rtdmpzorts&rt½mp[]_q]u¥m/kpuv°5_ºrt¬ mp[‡_Œ5rd_5¯'Œ uv_5dmokªuv½mp[‡_ kp_Gwdx]_5ˆŒ _t¶ ÌjÐ Ÿ Ì     R < <E0 LFJH % H RHXL HIF StHa(A, B) JE O (pq M (pτ )) L Oe (p qτ )  L LM L (H (A, B)) = O(pτ ) €¾_5mÉmp[]_w‰x‡r|mpuv_5dm¾q+rjr|m¾mo[‡„;mŒ5rtzozp_PkŠ†+rt‡‹‡k+mpr §;qˆ_ StHaQ(A, B) = (Q , Q , . . . , Q , H ) ¶ B) Z []_Bjx]^"qˆ_Gzr|¬8Œ5rj_ ¯'Œ uv_5dmok uv StHaQ(A,StHa(A, v u k t „ ‡‹ºmp[]_Gu¥z q‡u¦mUkpu¥°G_7uvk O(pτ ) ¿ÁŒ|¶ 3¬ $ Q‰G§ P'&¹À.¶ B) O(q) ÌjÐ Ÿ Ì      G RLH E H<

(34) L LH H G  FFNHIE H E 0H G gcd L A E  B E

(35)  L HI JE O (q lgqM L Oe (p q τ ) (pτ )) ÌjÐ Ÿ Ì   !  R  WE  LFJH % H RH L H  H G MNRH LE L StHa(A, B) LM  š. ]N^. B. 2. r. ›. 'r. s. . r. q. q. j. s. ?q. kj. 0. š. ]N^. kj. r. ›Wœ. r. r. B. r. r. . Mq. p. B. r. q. q. h. 1. k−1. k. r. s. B. ER

(36)  a r uwqR ar ∈ Q∪{±∞} E SqG5

(37) JH F#"+ H L'FH σ r uTJHq  L j UlPJH9n O (q lg qM (max (pτ, qσ))) L O (qM (max (pτ, qσ)))  h StHaQ(A, B)  SN n  LkjHI  s%$ E

(38) LHq@BH?qRDL j U-lYJH9n   š›œ. ]N^. . Mq. q. ?q. j. q. h. B. eB (q max (pτ, qσ)) s O. ·¸ý^/„|ji ŒG„tkp_GkG§_‰¶ šˆ¶Nzo_G„ts zordrtm'uvkprts=„;mou¥r‰¾§ kŠuvšt4_5–;„ts¥x‡„|mpuvrt¾§Œ5rt^`†‡„|zou=kŠr‰ rt¬B„ts¥š‰_5q]z)„|u=Œjx]^bq+_5z)k5§&­&_ ]_G_G‹mo[]_"_G–;„|svx‡„;mou¥r‰µr|¬ StHa(A, A ) r;–t_Gz~„ºzo„|mpuvrt‡„tsjx]^bq+_5z>r|¬@q‡u¦mBkŠuv°5_ O(pτ ) ¶7·³¬L­&_b†+_5zp¬¹rtzo^mp[]_ _5–;„ts¥x‡„|mpuvrtqjiB² r‰zp‡_5zPÒ k0zpx‡s¥_Lmp[]_G¬¹rtz8_5–‰_5zoi7†ˆr‰s¥ij]r‰^ªu=„|sduvbkŠ_Pwdx]_5‡Œ5_t§Ömo[]_5zo_„tzp_ Ω(p) §|­_^bx‡kŠm†ˆ_GzŠ¬¹r‰zp^ ^bx‡s¦mou¥†]svu=Œ5„;mou¥r‰‡k q+_ mn­&_5_Gµjx]^bq+_5z)k rt¬Tq]u¦m7kŠuv°5_ O(pτ ) „t‡‹ O(p τ ) §+mp[jx‡kmo[]_br;–‰_5z)„|svsŒ r‰^`†]s¥_5¨lu¦mni Ω(p) u=k O (p M (pτ )) ¶ ² r;­&_5–‰_5zP§‰­[‡_5­_BŒ r‰^`†]xlmp_>mo[]_Œ r‰^ª†‡s¥_5mp_ uv ¿ÁZ [¾G¶ OdÀ.§]mp[]_Bwdx]r|mou¥_Gdm q+rdrtm~u=k7Œ rt^`†]x]mp_G‹µu¥^`†]svuvŒ5u¦mos¥0i $ P]§ Q&F¶7Z [jx‡k5§W­&_"StHa(A, ŒG„|µx‡kp_Amo[]_`) w‰x‡r|Ompuv_5(pdm>Mqˆrj(pτ r|m>u¥))rtz)‹l_5zUmpr©†ˆ_GzŠ¬¹r‰zp^mp[‡_ _5–;„ts¥x‡„|mpuvrt_5–‰_5u¦¬T­&_B[‡„Ö–t_7„|svzo_G„t‹]iºŒ r‰^ª†‡xlmp_P‹„ts¥s¾mp[]_†+rtsvij]rt^`u=„|s=k u¥mo[]_"admox]zp^ª±¸²U„tq]uvŒ)[dmUkp_Gwdx]_G‡Œ _‰¶  rtmpu=Œ _7„|s=kŠrbmp[‡„|m mp[]_7Œ r‰^ª†‡xlmo„|mpuvrt©kp[]r‰x]sv‹ºqˆ_7kŠmo„|zpmp_P‹'qji`mp[‡_~s=„tkŠm&_Gs¥_G^`_5dm&rt¬¾mp[‡_7wdx]rtmpuv_5dm qˆrjr|mkprª„‰k mpr/„Ö–‰rtu=‹/mp[‡_Œ r‰kŠmpsviºŒ5rt^`†]xlm)„;mpuvrtrt¬mn­&rª†+rtsvij]rt^`u=„|sÉ_G–Ö„ts¥xˆ„;mpuvrtˆkxˆkŠuv]š/² r‰zp]_GzGÒ kkoŒ)[]_5^`_t¶ !@–‰_5/mo[]rtx]š‰[`mp[]u=k „|†]†‡zprd„tŒ)[`uvkrt†]mpuv^`„tsY§du¥muvd–‰rtsv–t_Pk@q]uvšbŒ r‰‡kŠmo„|dm)k@uvºu¥mok&Œ rt^`†]sv_ ¨lu¥mnit§‰mp[jx‡ku¦m&uvk]rtm _ ¯'Œ5u¥_G‰m7uv«†]z)„tŒ mpu=Œ _b­[‡_5µmo[]_ªs¥_G]š|mo[µr|¬@mo[]_`kŠ_Pw‰x‡_5‡Œ5_bu=k~‡r|mBkŠxl¯'Œ5u¥_G‰mos¥iq]uvšrtz>­[]_5«mp[]_`kŠ_Pwdx]_5‡Œ5_ u=kB‹l_ ¬¹_PŒ.mpuv–t_‰¶/e”rtzo_5r;–‰_5zP§+kp†ˆ_PŒ u=„|smo_GŒ)[]]u=wdx]_GkBkp[]r‰x]sv‹«qˆ_/xˆkŠ_P‹µ¬¹r‰zBu¦m)k7u¥^`†]sv_5^`_5dm)„;mpuvrtmpr”„Ö–‰rtu=‹«Œ rdknmos¥i rt†+_5z)„;mou¥r‰‡k ­u¥mp[”z)„;mpuvrtˆ„|sɍjx]^bq+_5z)kG¶&ajrˆ§‡„tku¥mUu=k „|sv­ „Öijk&mo[]_Œ5„‰kŠ_B­u¥mp[”r‰†lmpuv^/„|s„ts¥š‰_5q]z)„|u=Œ7„|svštr‰zpu¥mp[]^/kG§ mp[‡_Bu¥^`†]sv_5^`_5dm)„;mpuvrtu=k ¬Á„|z ¬¹zort^1„"mozpuv–juv„tsWmo„tkp¼W¶ 0. 2. B. 3. 0. B. 2.

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(41) . ÌjÐ Ÿ Ì  . MqR 0pR oh jRFHLh L Oe (p2 τ ) r E  L (A (p lg pM (pτ )) š . ]N^.  . A r s s Ared r red ) = O(p + τ ) s. E

(42)  LkjHI +hL. 0. StHa(A, A ) r. JE. ¾€ _5m W (a) ‹l_5‡r|mp_7mo[]_jx]^"qˆ_Gz r|¬^`rj‹]u‡_P‹”kŠuvšt”Œ)[‡„t]št_Pk r|¬8mp[]_b_5–;„|svx‡„;mou¥r‰r|¬ StHa(A, B) r;–‰_5z ¶  rtmpu=Œ _~mp[‡„|m ‹lrj_Gk&]r|m zo_ ¬¹_5z&mormp[‡_>xˆkŠx‡„tsɌ rtx‡‰mou¥‡šbr|¬8kŠuvšt©–Ö„tzpu=„;mou¥r‰‡kG§dkpu¥ˆŒ _~kp†+_GŒ u=„|sɌG„|zo_ a kp[]rtx]s=‹©qˆ_7m)„|¼t_GºW¬¹r‰z mp[](a)_B†‡zp_PkŠ_G‡Œ _7r|¬Œ r‰‡kŠ_PŒ xlmou¥–‰_7°5_5zor‰k$ Qt§ QP‘'&F¶ ÌjÐ Ÿ Ì      H A, B ∈ Z[X]

(43) WUH JM' FJ RL E L<  G A    R   OB. B. (A,B). (A,B). š. ]N^. f. .  E  A   H?qR Pr FJM'r H JM Lh A s JE (a, b) r HqGE W (a) − W 0. Ð#Ÿ ÐL ' 'YL lHŸ Í L  h. (A,B). $ JE B = A h. A. 0. HqGE. $.  

(44) LH?q E LE I  L LH Lh on. h. j. j. a<b P A 0 s (b) = sign (A (γ)B(γ)) (A,B) γ 0. StHa(A, A ). on. E . uwq. γ. 0p. oh. E r F LM' HqG<L+LH  Lh.  BH?qR  H F <pRE E  Mq. (a, b) s. s. A.  LERH  HqGBE W

(45) BLh.

(46)  -©¾(0 -   . ·¸»mp[]u=kkprtsv–t_GzG§¾­&_`x‡kp_`mp[]_/zo_5†]zo_Gkp_5dm)„;mpuvrtr|¬ †ˆr‰s¥ij]r‰^`uv„tsvk7uv»mp[]_'´&_5zo‡kŠmp_Gu¥»q‡„tkpu=k5¶'‡rtz a < b ∈ R § ¿ i = 0, . . . , dÀLmp[]_7´&_5zo‡knmo_5uv'q‡„‰kŠu=kLrt¬ R[x] rt©u¥dmp_Gzp–;„ts ­&_~‹l_G]r|mo_~qji B (x; a, b) =  ¶ [a, b] Â]r‰z~„tdi†+rtsvid‡rt^`uv„ts f ∈ R[x] = P b B (x; a, b) §+mp[]_ªŒ rj_ ¯'Œ uv_5dm)k b = (b ) „tzp_bŒG„|svs¥_P‹ mp[‡_ LERH LL  KERH 8rt¬ f ¶ £ _U‹l_5‡r|mp_qji V (f, [a, b]) §tmp[]_Udx‡^bq+_5z@rt¬Ékpuvšt`Œ)[‡„|]š‰_Gkuv`mp[]u=kLkp_Gwdx]_G‡Œ _ ¿Ázp_G^ªr;–juv]š"mp[]_ À ¶ b Z []_ª¬¹rtsvsvr;­u¥]š0mo[]_5r‰zp_G^§0­[]u=Œ)[u=k„‹luvzp_PŒ.mbŒ r‰‡kp_Gwdx]_5ˆŒ _`r|¬ ‚U_PkpŒG„|zpmp_GkGÒÉzox]s¥_‰§0„ts¥svr;­ k>x‡k7mpr”q+rtx‡‡‹ mp[‡_Bdx‡^bq+_5zr|¬zo_G„|sWzorjr|mok rt¬ f rt©mp[]_Buvdmp_Gzp–;„|s [a, b] ¿Fadmp_GZ† ‡¶ „ªrt¬0mo[]_Buvkprts=„;mou¥r‰©„ts¥š‰rtzou¦mo[]^'FÀ ! 6 Ÿ Ð Ð + *¹+ž * Ð ,    RE W

(47)  N L     L LH<L f LE (a, b)  <

(48) LE P 

(49) V (b) E  . RM N ≡ V (f, [a, b]) mod 2 Z []_”kp†]s¥u¥mŠmou¥‡šajmp_5K† ]„t¶ ‹½‡‹ u=k`q‡„‰kŠ_P‹½rt ‹l_ "&„tkŠmp_Gs {Š„tx¾Ò k`„ts¥š‰rtzou¦mo[]^§@­[]u=Œ)[D†]zorlŒ _5_P‹]k`„tk"¬¹rtsvsvr;­ Dk $ Q|§L—t'’ & ! b = b , i = 0, . . . , d, ¿Y—tÀ + t b (t), 0 ≤ i ≤ d − r, 0 ≤ r ≤ d. b = (1 − t) b ·³mº„|svsvr;­ k"xˆkªmpr½Œ r‰^`†]xlmp_mo[]_”zp_G†]zo_Gkp_5dmo„|mpuvrt½rt¬ f ¬¹rtz`mp[‡_mn­r®kŠx‡q]u¥dmo_5zo–Ö„tsvk [a, (1 − t)a + tb] „t‡‹ ¶ U„|^`_Gs¥i‰§ ¿Ázp_PkŠ†¾¶ À„|zo_>mp[]_"Œ rtdmozpr‰s¾Œ rj_ ¯'Œ uv_5dm)k [(1 − t)a + tb, b] r|¬ f r‰ [a, (1 − t)a + tb] ¿Ázpb_PkŠ†¾¶ =[(1(b−) t)a + tb, b]À ¶ b = (b ) Z []_7r|mo[]_5z kŠmp_G†‡k „tzp_7kŠuv^`u¥s=„|z morªmp[]_badmox]zo^"±¸²U„tq]u=Œ)[‰m kprtsv–t_GzG¶ d (x−a)i (b−x)d−1 i (b−a)d. i d. d i=0 i. d. _. ˆ\. š . d. q. i d. i i=0,...,d. h. h. 0n. u. q. s. 0 i. i. i 0 i=0,...,d. −. . €. r−1 i+1. r−1 i. r i.      ! "$# E‡. ”. ƒ. ”. Œ2“Xƒ ”. Œ2„. d−i i=0,...,d i. +. ". “XƒW”. £ _ ‡zokŠm zo_GŒ5„ts¥s¾kprt^`_B†+rtsvij]rt^`u=„|sWmpz)„|ˆkn¬¹r‰zp^/„;mou¥r‰‡k&zp_Gsv„|mp_P‹'mor`mp[]_b´&_Gzp‡kŠmp_Gu¥zo_5†]zo_Gkp_5dmo„|mpuvrt $ —t’&³¶L€¾_5m q+_©mp[]_”kp_ m`r|¬U[]r‰^`rtšt_G]_5r‰x‡k"†ˆr‰s¥ij]r‰^ªu=„|s=k"r|¬~‹l_Gštzo_5_ u¥ ¶ ‡rtz`„|ji §@­&_ R[x, ‹l_G]r|y]mo_bqji p mp[‡_b[]r‰^ªr‰št_G]uvko„;mou¥r‰”r|¬ p uv«‹l_Gštzo_5_ d ¶7Â]rtz λ 6= 0,dµ ∈ R(x,§ÉŒ y)rtˆkŠu=‹l_5zUmp[]_b¬¹prtsvs¥∈r;­R[x] uv]š'^/„t†‡k ! R →R % ρ : (x, y) 7→ (y, x)§ % H : (x, y) 7→ (λx, y) § H : (x, y) 7→ (x, λy) § % T : (x, y) 7→ (x − µy, y)§ T : (x, y) 7→ (x, y − µx)¶ Z []_Gu¥zTŒ r‰^`†ˆrdkŠu¥mpuvrt­u¥mp[ u¥‡‹]x‡Œ _ uvd–‰_5zpmpuvq]sv_@mpz)„|ˆkn¬¹r‰zp^/„;mou¥r‰‡k0rtmo[]_ kŠ_5m8rt¬ˆ[]r‰^ªr‰št_G]_5r‰x‡k¾†+rtsvij]rt^`u=„|s=k r|¬‹l_Gštzo_5_ d §0­[]uvŒ)[Œ5rtzozp_PpkŠ†+rt‡‹µmprmp[‡_`¬¹rtsvs¥r;­uv]š^/„|†‡kB‹l_G]r|mo_G‹»­u¦mo[»mp[]_'ko„|^`_`‡„|^`Y_ ! ∀p ∈ R[x] § § § § § ¶ ) ρ(p) = x p(1/x) H (p) = p(λx) H (p) = p(λ x) T (p) = p(x − µ) T (p) = (1 − µ x) p( d. [d]. 2. 2. 0 λ. λ. 0 µ. µ. d. d. λ. 0 λ. −1. µ. 0 µ. d. x 1−µ x.

(50) ‘. 5<JF  LFJEVF YF P. Â]r‰z„|ji'†+rtsvid‡rt^`uv„tsY§ p(x) = P. d i i=0 bi Bd (x; a, b). §]­&_>[ˆ„Ö–t_. ρ ◦ T1 ◦ ρ ◦ Hb−a ◦ T−a (p) =. P Â]r‰zU„|‡r|mp[‡_5zUuvdmp_Gzp–;„|s § mpr P b x u=k [c, d] p(x) = d d i=0 i. d 0 i i=0 bi Bd (x; c, d). 0 i i. d   X d. i. b i xi .. „|‡‹mo[]_^/„|†”­[‡uvŒ)[mozo„t‡kn¬¹r‰zp^/k P i=0.  i d d i=0 i bi x. ¿ dÀ ·³¬ [a, b] = [0, 1] „t‡‹ [c, d] = [0, ] §t^/„t†¿ ‰À0qˆ_PŒ r‰^ª_Pk! ρ◦T ◦ρ◦H ◦ρ◦T ◦ρ ¶0 ¬ mp_GzLkŠuv^`†]svuˆŒG„;mou¥r‰‡k5§ ­&_>r‰qlmo„tu¥ ¿ OjÀ y y ∆ : p 7→ p(x + , ) = p ◦ T ◦ H . e”x]s¥mpuv†]sviju¥]š`mo[]_B†+rtsvid‡rt^`uv„tsWqdi 2 ijuv_5s=‹]k mp[‡_7¬¹rt2svs¥r;2­uv]šª^`„t† ∆ : p 7→ p(2 x + y, y) ­[]u=Œ)[rt†+_5z)„;mp_Pk rt†+rtsvid‡rt^`uv„tsvk ­u¥mp[uvdmp_5š‰_5z Œ5rd_5¯'Œ uv_5dmokG¶ ·³¬ „|ˆ‹ §8^/„|† ¿‰À7qˆ_PŒ r‰^ª_Pk! ¶·³m Œ r‰zpzo_Gkp[a,†+rt‡b]‹]k=mor"[0,mo[]1]_7¬¹rtsvsvr;­[c,u¥]d]šª^/=„|†[ rt, ©1]mp[‡_B[]rt^`rtš‰_5]_Grtx‡k&†+rtsvρij]◦rtT^`u=„|s=◦k ! ρ ◦ H ◦ T ◦ ρ ◦ T ◦ ρ 0 1 1 ρ ◦ T1 ◦ ρ ◦ Hd−c ◦ T−c ◦ Ta ◦ H b−a ◦ ρ ◦ T−1 ◦ ρ = T10 ◦ Hd−c ◦ Ta−c ◦ H b−a ◦ T−1 1 2. −1. −. 1. 1 2. 0. −1. d. 1 2. −. 1 2. −1. 1 2. 1. − 12. x x 0 ∆+ : p 7→ p( , + y) = p ◦ T−1 ◦ H 21 . 2 2.  šd„|uv¾§¾^"x]s¦mou¥†‡s¥ijuv]šqdi iju¥_Gsv‹‡k>mp[]_`^/„|† ­u¥mp[u¥dmo_5št_Gz Œ rj_ ¯'Œ5u¥_G‰m)k52¶ 6 Ÿ Ð Ð + *¹+ž * Ð , H (b ) ∈ Z d. ∆+ : p 7→ p(x, x + 2 y). ­[‡uvŒ)[»rt†+_5z)„;mo_Gk~rt»†ˆr‰s¥ij]r‰^ªu=„|s=k. __ f ˆ\ JE H?qRDC FERHIJE d+1

(51)  H qG  L+ E H Lh  jGLonEL  p 

(52) 'F SLE@HqG JERHIFM [a, ib]i=0,...,d 

(53) S  

(54)  L   E   L @ E qGJ F#"+ H MqG=LkjUlYJH n Lh  L j H JE% HqG    E  U    H s C FERHIJE L  KERH SLh pr hL H?qRH9τu.LZFR

(55) JERHIFM  [a, a+b ] r [ a+b  S

(56) LE Y 

(57) /n Oe (d(τ + d)) B 2 2 , b] E H qGJF# "+: 

(58) LE P S

(59) 0n O (τ + d) s 6 Ÿ оÐx ¢ U / kpu¥‡šbmo[]_>‹]_&" „‰knmo_5s {Š„|x©koŒ)[]_G^ª_ª¿Y—‰ÀU¿¹¬¹rtz t = 1 À.§j­_>†]zor;–t_ qji/uv‡‹lx‡Œ mpuvrtºmp[‡„|mmo[]_BŒ rj_ ¯'Œ uv_5dm)k. t„ zp_r|¬~mp[]_¬¹r‰zp^ §L­[‡_5zo_ u=k`r|¬>kpuv°5_ ¶ýƒ_G‹]x‡Œ uv]š»mpr»mp[‡_µkp„t^ª_ ‹l_G]=rt^`uv‡„;mortz 2 §l­&_Brtqlm)„|uv©uvdmp_5š‰_5zŒ rj_5¯/Œ5u¥_Gdmok br|¬T∈kpu¥°GZ_ ≤ τ + d ¶ ≤ τ + r £ _7‹]_5]rtmp_7qji mp[‡_kŠuv°5_7r|¬0mp[]_Œ5rj_ ¯'Œ uv_5dmok ­[‡_5zo_ „tzp_~mp[‡_Œ rj_ ¯'Œ uv_5dm)k r|¬ f uvmp[]_´&_Gzp‡kŠmpτ_Gu¥q‡„‰kŠu=k (B (x; a, b)) ¶ ( rtmpbu=Œ )_>mp[ˆ„;m τ ≤ τ +(bd ¶) Â]r‰z>Œ5rt^`†]xlmou¥‡šºmo[]_ªŒ5rj_ ¯'Œ uv_5dmok>rt¬ rt „|‡‹ §É­_ª„|†]†]svimp[‡_`ko„|^`_"rt†+_5z)„;mpuvrtˆk~„‰k ­[]_G/­_ Œ5rt^`†]xlmo_ mp[]_UŒ5rj_ ¯'Œ uv_5dmok@r|¬É„7f†ˆr‰s¥ij][a,r‰^ªu=„|s]]¬¹rtzTmo[]_ [ ´&_5zo,‡b]knmo_5uv`q‡„tkp_GkTr‰ [0, ] „|‡‹ [ , 1] §t­[]_G u¥m uvk š‰u¥–‰_5u¥mp[‡_´_Gzpˆknmo_5uv©q‡„‰kŠu=k rt [0, 1] ¶ Œ5Œ5rtz)‹luv]šµmor ¿ OdÀ5§„|†‡†]s¥ijuv]šmp[]_”‹l_ "&„tkŠmp_Gs {Š„tx „ts¥š‰rtzou¦mo[]^ Œ rtzozo_Gkp†ˆr‰‡‹]$k ‡zokŠm`mpr»^bx]s¥mpuv†]svi½qjimp[‡_ q]uv]rt^`u=„|s Œ5rd_5¯'Œ uv_5dmokG§@mo[]_5 mprkp[]u¥¬ m y → x + y §mp[‡_5 mpr½koŒ5„|sv_rt]_–;„|zouv„tq]sv_©rt¬Ump[‡_”[]rt^`rtš‰_5]_Grtx‡k †+rtsvid‡rt^`uv„ts p qdi §]„t‡‹ ‡‡„ts¥svi/mpr/‹luv–juv‹]_Bqdi'mp[]_Bq]uv]r‰^ªu=„|s0Œ rj_ ¯'Œ uv_5dm)k P¶ ajuv‡Œ _ªmo[]_'kŠuv°5_/rt¬Lmo[]_/q]uv]rt^`u=„|sLŒ5rd_5¯'Œ uv_5dmok7u=k7qˆr‰x]‡‹l_P‹»qdi ¿ mo[]_5uvzbkpx]^¤u=k À §¾mo[]_'Œ rdknmBr|¬&mp[‡_ ‡z)knmBkŠmp_5†«uvk>q+rtx]‡‹]_G‹µqji Oe (d(τ + d)) ¶Z [‡_ªkp[]u¥¬ m7zo_Gwdx]uvzo_Gk Oe d(dτ ) q]u¥m7rt†+_5z)„;mp2uvrtˆk $ t’l§ÉZ [¾¶É’‡U¶ QÖ' &³¶ ajuv‡Œ _>mo[]_>kpuv°5_>r|¬mo[]_Gkp_7Œ rj_ ¯'Œ uv_5dm)ku=k&qˆr‰x]‡‹l_P‹ºqji O(τ + d) §]koŒ5„ts¥uv]šª„b–;„|zou=„|q]sv_~qji „t‡‹ºŒ5rt^`†]xlmou¥‡š mp[‡_w‰x‡r|mpuv_5dmqji'mo[]_Bq]uv]rt^`u=„|s¾Œ rj_5¯/Œ5u¥_Gdmok zo_Gwdx]uvzp_Pk Oe (d(τ + d)) q]u¦mp±Fr‰†ˆ_Gzo„|mpuvrt‡kG¶ Z []_Gzp_5¬¹rtzo_t§mo[]_Œ rt^`†]sv_ ¨lu¥mni4r|¬ªŒ5rt^`†]xlmou¥‡šDmo[]_»´&_Gzp‡kŠmp_Gu¥ Œ rj_ ¯'Œ5u¥_G‰m)kr|¬ f r‰ mo[]_kŠx]q‡u¥dmp_Gzp–;„ts uvk&q+rtx]‡‹]_G‹'qji ¶@´i/kpij^ª^`_5mpzoit§ju¥j–‰_5zpmpuv]šBmp[‡_~r‰zo‹l_GzLrt¬Émp[]_7Œ rj_ ¯'Œ5u¥_G‰m)kLrt¬ f §‰­&_ [a, rtq]mo„|uv©]mp[]_ko„|^`_7qˆr‰x]‡‹©Oe¬¹rtz (d(τmo[]_+Œ rjd))_5¯/Œ5u¥_Gdmok r|¬ f r‰ [ , b] §‡­[]uvŒ)[_5ˆ‹]k mp[‡_B†]zprjrt¬n¶ 2 0Ú)߁çÖänänåÖänådò|Õ ðlÙÉäÔ;Þ äTßFÚÞ)ÝPÝÖá âUÛYänÝ|äŠÞoßFá â~ßFÔÖäT×YÔÖÕ ð¦ß¾Ú.ÝtänÛYÞoßFÕØÚ.ç (br−1 +br−1 i i+1 ) 2 d. bri. 2. r. r i. bi 2i. d i. 0. i d. i i=0,...,d. i i=0,...,d. 0. i=0,...,d. a+b 2. a+b 2. 1 2. . 1 2. n. B. B. B. . a+b 2. 1 2. B. a+b 2. 0. 1 2.

(60) ’.   

(61) 

(62) . ! "$# &%')(+*,-©b.*/   8®/7(  0 7(05·¸'mo[]u=kkŠ_PŒ.mpuvrt§d­&_>qˆr‰x]‡‹/mo[]_7jx]^bq+_5z r|¬0q]u¦mr‰†ˆ_Gzo„|mpuvrt‡k@¬¹rtz&mp[]_7u=kŠr‰sv„|mo„tu¥r‰/r|¬zo_G„ts+zorjr|m)k5§dqjiºadmpx]zo^Ò k „|ˆ‹©´&_5z)kp‰mo_5uv¾Ò k ^ª_5mp[]rl‹É¶ £ _Œ r‰‡kpuv‹l_Gz&mp[]_7mozp_G_7„‰kpkprlŒ u=„;mo_G‹'­u¥mp[”„ªzox]r|¬mo[]_kpx]q+‹]u¥–ju=kŠuvrt„|svštr‰zpu¥mp[]^ rt½„µ†ˆr‰s¥ij]r‰^ªu=„|s f ¶ !L„tŒ)[®]rl‹l_©zp_G†]zp_PkŠ_GdmokB„|uvdmp_Gzp–;„|sF¶”Z []_ºzorjr|mrt¬&mo[]_'mozp_G_'Œ r‰zpzo_Gkp†ˆr‰‡‹]k7mor”mp[‡_ uv]u¦mouv„ts u¥dmp_Gzp–;„ts ¶ !L„‰Œ)[Duvdmp_5zo–;„|s&­[]u=Œ)[ u=kª]r|m/„«sv_G„Ö–‰_'r|¬~mp[]_©mpzo_5_u=kªkp†]svu¦m/uvdmprmn­r[‡„|s¥¬ uv‰mo_5zo–;„|s=k5¶LZ [‡_‹lI_5†lmo=[”[a,r|¬Tb]„`]rl‹l_rt¬0mo[]_7mpzo_5_'¿Y„tkokŠrlŒ u=„;mo_G‹©­u¦mo[µ„|uv‰mo_5zo–;„|s I À&uvk lg(|I |/|I|) ¶L·³mUu=k „|s=kpr mp[‡_Bdx‡^bq+_5zr|¬Tkpx]q+‹]u¥–ju=kŠuvrt†ˆ_GzŠ¬¹r‰zp^`_P‹'mor/rtq‡„|mpuvºmo[]_kŠx‡q]u¥dmo_5zo–Ö„ts I r|¬ I ¶ 0. 0.   . S€. ‚S•S„kŒH†ˆ” <† ”)”.   . <Œ2‘ƒ E†ˆ“ŒHƒ  E–. 0. . “Xƒ ”. . ¶ £Z []_º_µ„tŒ5kortkŠx‡^`^ª†]_`xlm)mo„;[‡mp„;uvrtm  dr|=¬ fO(τ )ŒG§T„|mp [jx‡qˆk _L‹lrt(f]_uv) =Oe O(τ(d )τ¶ ) U¿ÁZ r|mp[¾u=Œ ¶_©“;À „|¶ s=kŠr mort[‡mpu=„;Œ m"_„|mp¬ [ˆmp_G„;zBm mpL[]u=(fkªŒ rt^`) †]=x]moO(d „;mou¥r‰¾+§0mpτ[])_ admox]zp^ª±¸²U„tq]uvŒ)[dmBkŠ_Pwdx]_5‡Œ5_ uvk7„Ö–;„|uvsv„tq]sv_t¶ £ _/‹lr]r|mB]_G_G‹”mo[]_/Œ r‰^ª†‡s¥_5mp_/kŠ_Pwdx]_5‡Œ5_"q‡xlmrt]svi mp[‡_7wdx]rtmpuv_5dm qˆrjr|mP§jmp[jx‡k&mo[]StHa(f uvkŒ5rt^`†])xlm)„;mou¥r‰©Œ5„t©q+_B‹lrt‡_~uv Oe (d τ ) ¿YZ [¾¶‡|À ¶@² r;­&_5–‰_5zP§‰­&_~^/„Öi'„tsvkpr „tkokpx]^`_ mp[‡„|mLmp[‡_UŒ r‰^ª†‡s¥_5mp_Ukp_Gwdx]_G‡Œ _ u=k@Œ5rt^`†]xlmo_G‹É§d­u¦mo[ºŒ r‰^`†]s¥_5¨lu¦mni Oe (d τ ) ¿YZ [¾¶POjÀ.§jkŠuv‡Œ5_mp[]u=kLkŠmp_G† u=k ]r|mmo[]_BqˆrtmŠmos¥_G]_GŒ)¼'rt¬0mo[]_„|svštr‰zpu¥mp[‡^¶ 2. B. red. red. red. B. 2. B. S€ . 3.      

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