Finding Exact Minimal Polynomial by Approximations

Finding Exact Minimal P olynomia l b y Appro xima tions ∗ Xiaolin Qin Y ong F eng Jingw ei Chen Jingzhong Zhang Lab orato ry for A utomated Reasoning and Programming Chengdu Institute of Computer Applications Chinese Academ y of Sciences 610041 Chengdu, P . R. Chin a Lab orato ry of Computer Reasoning and T rust w orthy Computation Univ ersity of Electronic Science and T ec hnology of China 610054 Chengdu, P . R. China E-mail: qinxl811028@163 .com, y ongfeng@casit.ac.cn Abstract W e present a new algorithm for reconstructing an exact algebraic num- b er from its approximate v alue using an impro v ed parameterized integer relation construction metho d. Our result is almost consisten t with the existence of error con trolling on obtaining an exact ratio nal num b er from its app roximation. The algorithm is app licable for finding exact minimal p olynomial b y its app roxima te root. This al so enables us t o pro v ide a n efficien t meth od of conv erting t he rational ap p roxima tion rep resentation to the minimal p olyn omial representation, and devise a simple algorithm to factor multiv ariate p olynomials with rational coefficients. Compared with other metho ds, th is metho d has the numerical compu - tation adv antage of high efficiency . The ex p erimental results show that the method is more efficien t than identify in Maple 11 for obtaining an exact algebrai c num b er from its approximation. In this pap er, we com- pletely implement how to obtain exact results by numerical approxima te computations. keyw ords: Algebraic num b er, Numerical approximate computa- tion, Symbolic-numerical computation, In teger relation algorithm, Mini- mal p olynomial. ∗ The wo rk i s partially s upported by China 973 Pr o j ect NKBRPC- 2004CB31800 3, the Kno wledge Inno v ation Program of the Chinese Academy of Sciences KJCX2-YW-S02, and the Nat ional Natural Science F oundation of China(Grant NO.10771205) 1 1 In tro duction Symbolic computatio ns are principa lly exa c t and sta ble . How ever, they hav e the dis adv antage of intermediate expr ession swell. Numerica l appr oximate com- putations can so lve larg e and complex pro blems fast, whereas only give ap- proximate results. The gr owing dema nd for sp eed, accura cy and reliability in mathematical computing has acceler ated the pro cess of blurr ing the distinction betw een tw o a r eas of resea rch that were previously quite separate. Therefore , algorithms that combine ideas from symbolic a nd numeric computations have bee n of increa sing interest in rec e nt tw o deca de s . Symbolic computations is for sake of s p ee d b y intermediate us e o f floa ting-p oint arithmetic. The work rep orted in [1, 2, 3, 4, 5 , 6] studies r ecov ery of appro ximate v alue from numer- ical intermediate results. A so mewhat rela ted topic are algor ithms that obtain the exact factoriza tion of a n exact input p o lynomial by use of flo ating p oint arithmetic in a practically efficient technique [7, 8 ]. In the meantime, symbolic metho ds ar e applied in the field of numerical computations for ill-co nditioned problems [9, 10, 11]. The main goa l of hybrid symbolic-numeric c o mputation is to extend the domain of efficient ly solv able problems. How ever, there is a gap betw een approximate computations a nd exact r esults[12]. W e consider the following question: Suppo se we are given a n approximate ro ot of a n unknown p o ly nomial with integral co efficients and a b ound on the degree and size of the co efficients of the p olyno mial. Is it po s sible to infer the p olynomia l a nd its exact r o ot? The question was rais ed by Man uel Blum in Theo retical Cryptogr aphy , and Jingzho ng Zha ng in Automated Reaso ning, resp ectively . Kannan et al a nswered the question in [13]. How e ver, their tech- nique is bas ed on the Lenstra-L e nstra-Lov asz(LLL) lattice r eduction alg orithm, which is quite unstable in numerical co mputations. The function MinimalPoly- nomial in maple , which finds minimal p oly no mial for an a pproximate ro ot, was implemen ted using the same technique. In this pap er, we present a ne w algo- rithm for finding ex act minimal p oly nomial and reconstructing the ex act ro ot by approximate v alue. Our algorithm is base d on the impr ov e d pa rameterized int eger relation construction algorithm, whose stabilit y admits an efficient im- plement ation with lo wer run times on a verage than the former algor ithm, and can be used to prov e that rela tion bounds obtained from computer runs using it are n umer ically accur a te. The other function id entify in maple , whic h finds a clo sed form fo r a decimal appr oximation o f a num ber, was implemented using the integer rela tion constr uctio n a lgorithm. How ever, the choice o f Digits of approximate v alue is fairly arbitra ry [14]. In contrast, we fully analyze numer- ical be havior of an approximate to exa c t v a lue a nd g ive how many D igits o f approximate v alue, which can b e obtained exact results. T he work is reg ard as a further research in [15]. W e solve the problem, whic h ca n b e describ ed as follows: Given approximate v alue ˜ α at ar bitrary a ccuracy of an unknown algebr aic 2 nu m ber , and w e also kno w the degree of the algebraic num ber n and an upp er bo und N of its height on minimal p olyno mial in adv ance. The problem will b e solved in tw o s teps: First, we discuss how m uch co ntrol er ror ε is, so that we can r e construct the alge br aic num b er α from its a pproximation ˜ α when it ho lds that | α − ˜ α | < ε . Of cours e, ε is a function in n a nd N . Second, we give an algorithm to co mpute the minimal p oly nomial of the algebraic num b er. W e a re able to extend o ur results with the same metho ds to devise a simple po lynomial-time algor ithm to factor multiv ariate po lynomials with ra tio nal co - efficients, and provide a natural, efficie nt technique to the minimal po lynomial representation. The rest of this pap er is org anized as follows. Section 2 illustrates the im- prov ed parameter ized integer rela tion cons truction algor ithm. In Section 3 , we discuss how to recover a quadratic alg ebraic num b er a nd reco nstruct minimal po lynomial b y appr oximation. Section 4 giv es some exper imental results. The final section concludes this paper. 2 Preliminaries In this s ection, we fir st give some notations, and a brief intro duction on integer relation problems. Then an impro ved par a meterized in teg er relation construc- tion algorithm is a ls o reviewed. 2.1 Notations Throughout this pap er, Z denotes the set of the integers, Q the set of the rationals, R the set o f the reals, O ( R n ) the corre s p onding system of ordinar y int egers, U ( n − 1 , R ) the g roup of unitar y matrice s over R , GL ( n, O ( R )) the group of unimo dular matrices with entries in the integers, col i B the i-th column of the matrix B. The ring of p olyno mials with integral co efficients will be denoted Z [ X ]. The content o f a p oly nomial p ( X ) in Z [ X ] is the grea test common diviso r of its co efficie nts. A p olyno mial in Z [ X ] is pri mitiv e if its conten t is 1. A po lynomial p ( X ) has deg r ee d if p ( X ) = P d i =0 p i X i with p d 6 = 0 . W e write deg ( p ) = d . The l eng th | p | of p ( X ) = P d i =0 p i X i is the Euclidean length of the vector ( p 0 , p 1 , · · · , p d ); the heig ht | p | ∞ of p ( X ) is the L ∞ -norm of the vector( p 0 , p 1 , · · · , p d ), so | p | ∞ = max 0 ≤ i ≤ d | p i | . An alg ebr aic numb er is a ro ot of a p olyno mial with integral co efficients. The min imal pol y nomi al of a n alg ebraic nu m ber α is the irreducible p olynomial in Z [ X ] satisfied by α . The minimal po lynomial is unique up to units in Z . The deg r ee and heig ht of an a lg ebraic nu m ber are the degree and heigh t, re s p e ctively , o f its minimal p o lynomial. 3 2.2 In teger relation algorithm There ex ists an integer relation amongst the num bers x 1 , x 2 , · · · , x n if ther e ar e int egers a 1 , a 2 , · · · , a n , not all zer o , such that P n i =1 a i x i = 0. F or the vector x = [ x 1 , x 2 , · · · , x n ] T , the no nzero v ector a ∈ Z n is an integer r e la tion for x if a · x = 0. Here are so me us eful definitions and theorems[16, 17]: Definition 1 ( M x ) Assume x = [ x 1 , x 2 , · · · , x n ] T ∈ R n has norm | x | =1. De- fine x ⊥ to b e the set of al l ve ctors in R n ortho gonal to x . L et O ( R n ) ∩ x ⊥ b e the discr ete latt ic e of inte gr al r elations for x . Define M x > 0 to b e the smal lest norm of any r elation for x in this la ttic e. Definition 2 ( H x ) Assume x = [ x 1 , x 2 , · · · , x n ] T ∈ R n has norm | x | =1. F ur- thermor e, supp ose that no c o or dinate entry of x is zero , i.e., x j 6 = 0 for 1 ≤ j ≤ n (otherwise x has an imme diate and obvious inte gr al r elation). F or 1 ≤ j ≤ n define the p artial sums s 2 j = X j ≤ k ≤ n x 2 k . Given s u ch a unit ve ctor x , define the n × ( n − 1) lower tr ap ezoidal matrix H x = ( h i,j ) by h i,j =      0 if 1 ≤ i < j ≤ n − 1 , s i +1 /s i if 1 ≤ i = j ≤ n − 1 , − x i x j / ( s j s j +1 ) if 1 ≤ j < i ≤ n. Note t hat h i,j is sc ale invariant. Definition 3 (Mo difie d Hermite r e duction) L et H b e a lower t r ap ezoidal matrix , with h i,j = 0 if j > i and h j,j 6 = 0 . Set D = I n , define the matrix D = ( d i,j ) ∈ GL ( n, O ( R )) r e cu r s ively as fol lows: F or i fr om 2 to n, and for j fr om i-1 to 1(step-1), set q = ni nt ( h i,j /h j,j ) ; then for k fr om 1 to j re plac e h i,k by h i,k − q h j,k , and for k fr om 1 to n r eplac e d i,k − q d j,k , wher e the function nint denotes a ne ar est inte ger function, e.g., nint(t)= ⌊ t + 1 / 2 ⌋ . Theorem 1 L et x 6 = 0 ∈ R n . Su pp ose that for any r elation m of x and for any matrix A ∈ GL ( n, O ( R )) t her e exists a un itary matrix Q ∈ U(n- 1) such that H = AH x Q is lower t r ap ezoidal and al l of the diagonal elements of H satisfy h j,j 6 = 0 . Then 1 max 1 ≤ j ≤ n − 1 | h j,j | = min 1 ≤ j ≤ n − 1 1 | h j,j | ≤ | m | . 4 Remark 1 The ine quality of The or em 1 offers an incr e asing lower b ound on the size of any p ossible r elation. The or em 1 c an b e use d with any algorithm that pr o duc es GL ( n, O ( R )) matric es. Any GL ( n, O ( R )) matrix A whatso ever c an b e put into The or em 1. Theorem 2 Assume r e al numb ers, n ≥ 2 , τ > 1 , γ > p 4 / 3 , and t hat 0 6 = x ∈ R n has O ( R ) i nte ger r elations. L et M x b e the le ast norm of r elations for x . Then P S L Q ( τ ) will fi nd some inte ger r elation for x in no mor e than  n 2  l og ( γ n − 1 M x ) l og τ iter ations. Theorem 3 L et M x b e the smal lest p ossible norm of any rel ation for x. L et m b e any r elation found by PSLQ( τ ). F or al l γ > p 4 / 3 fo r r e al ve ctors | m | ≤ γ n − 2 M x . Remark 2 F or n =2, The or em 3 pr oves t hat any r elation 0 6 = m ∈ O ( R 2 ) found has norm | m | = M x . In other wor ds, P S LQ ( τ ) finds a shortest r elation. F or r e al numb ers this c orr esp onds to the c ase of t he Euclid e an algorithm. Based on the theorems as ab ove, and if ther e exists a known erro r co ntrolling ε , then an alg orithm fo r obta ining the integer rela tion was desig ned as follows: Algorithm 1 Par ameterize d inte ger r elation c onstruction algorithm Input: a ve ctor x , and an err or c ontr ol ε > 0 ; Output: an inte ger r elation m . Step 1: Set i := 1 , τ := 2 / √ 3 , and un itize the ve ctor x to ¯ x ; Step 2: Set H ¯ x by definition 2; Step 3: Pr o duc e matrix D ∈ GL ( n, O ( R )) using mo difie d Hermite Re- duction; Step 4: Set ¯ x := ¯ x · D − 1 , H := D · H , A := D · A, B := B · D − 1 , c ase 1: if ¯ x j = 0 , then m := col j B , goto Step 11; c ase 2: if h i,i < ε , then m := col n − 1 B , goto Step 11; Step 5: i := i + 1 ; Step 6: Cho ose an inte ger r, such that τ r | h r,r | ≥ τ j | h i,i | , for al l 1 ≤ j ≤ n − 1 ; 5 Step 7: Define α := h r,r , β := h r +1 ,r , λ := h r +1 ,r +1 , σ := p β 2 + λ 2 ; Step 8: Change h r to h r +1 , and define the p ermutation matrix R ; Step 9: Set ¯ x := ¯ x · R , H := R · H , A := R · A , B := B · R , if i=n-1, then goto St ep 4; Step 10: Define Q := ( q i,j ) ∈ U ( n − 1 , R ) , H := H · Q , goto Step 4; Step 11: r eturn m . By algor ithm 1, w e can find the integer relation m of the vector x = (1 , ˜ α, ˜ α 2 , · · · , ˜ α n ). So, we get a no nzero p olynomial o f degree n , i.e., G ( x ) = m · (1 , x, x 2 , · · · , x n ) T . (1) Our ma in task is to show that polynomial (1) is uniquely determined under assumptions, and discuss the controlling erro r ε in algo r ithm 1 in the next section. 3 Reconstructing minimal p olynomial from its appro ximation In this section, we will so lve such a problem: F or a given floa ting num b er ˜ α , which is an approximation of unknown algebra ic num ber , how do we obtain the exa ct v a lue? Without loss of genera lity , we first consider the recovering quadratic algebr aic num ber from its approximate v alue, and then gener alize the results to the case of algebra ic n umber o f high degree. A t fir st, we ha ve some lemmas as follows: Lemma 1 L et f = P n i =0 a i x i ∈ Z [ x ] b e a p olynomial of de gr e e n > 0 , and let ε = ma x 1 ≤ i ≤ n | α i − ˜ α i | for t he r est of this p ap er, wher e ˜ α i for 1 ≤ i ≤ n ar e t he r ational appr oximations to the p owers α i of algebr aic numb er α , and ˜ α 0 = 1 . Then | f ( α ) − f ( ˜ α ) | ≤ ε · n · | f | ∞ . (2) Pro of: Since f ( α ) − f ( ˜ α ) = P n i =0 a i ( α i − ˜ α i ), we get | f ( α ) − f ( ˜ α ) | = | P n i =1 a i ( α i − ˜ α i ) | , a nd then | n X i =1 a i ( α i − ˜ α i ) | ≤ n X i =1 | a i | · | ( α i − ˜ α i ) | ≤ n X i =1 | a i | · ε ≤ n · | f | ∞ · ε. So, the lemma is finished. 6 Lemma 2 L et h and g b e nonzer o p olynomials in Z [ x ] of de gr e e n and m , r esp e ctively, and let α ∈ R b e a zer o of h with | α | ≤ 1 . If h is irr e ducible and g ( α ) 6 = 0 , then | g ( α ) | ≥ n − 1 · | h | − m · | g | 1 − n . (3) Pro of: See Prop osition(1.6 ) o f[13]. If | α | > 1, a simple tra nsform o f it do es . Corollary 1 L et h and g b e nonzer o p olynomials in Z [ x ] of de gr e es n and m , r esp e ctively, and let α ∈ R b e a zer o of h with | α | ≤ 1 . If h is irr e ducible and g ( α ) 6 = 0 , then | g ( α ) | ≥ n − 1 · ( n + 1) − m 2 · ( m + 1) 1 − n 2 · | h | − m ∞ · | g | 1 − n ∞ . (4) Pro of: Fir st no tice that | f | 2 ≤ ( n + 1) · | f | 2 ∞ holds for any po lynomial f of degree at most n > 0, so | f | ≤ √ n + 1 · | f | ∞ . F rom Lemma 2 we g et | g ( α ) | ≥ n − 1 · ( n + 1) − m 2 · ( m + 1) 1 − n 2 · | h | − m ∞ · | g | 1 − n ∞ . So, the c o rollar y is finished. Theorem 4 L et an appr oximate value ˜ α b elong to an unknown algebr aic num- b er α of de gr e e n > 0 . Assume that the existenc e of the p olynomial G ( x ) = P n i =0 a i x i , wher e a n 6 = 0 . Su pp ose n and upp er b ound N on the de gr e e and height of minimal p olynomial g ( x ) on the algebr aic nu mb er α ar e known, r e- sp e ctively. If | G ( ˜ α ) | < n − 1 · ( n + 1) − n + 1 2 · | G | − n ∞ · N 1 − n − n · ε · | G | ∞ , then | G ( α ) | < n − 1 · ( n + 1) − n + 1 2 · | G | − n ∞ · N 1 − n . Pro of: F rom lemma 1, we notice that | G ( α ) − G ( ˜ α ) | ≤ ε · n · | G | ∞ , a nd | G ( α ) − G ( ˜ α ) | ≥ | G ( α ) | − | G ( ˜ α ) | , so | G ( α ) | ≤ | G ( ˜ α ) | + n · ε · | G | ∞ . F rom the assumption of the theor em, since | G ( ˜ α ) | < n − 1 · ( n + 1) − n + 1 2 · | G | − n ∞ · N 1 − n − n · ε · | G | ∞ . (5) So, the theory is finished. Corollary 2 If | G ( α ) | < n − 1 · ( n + 1) − n + 1 2 · | G | − n ∞ · N 1 − n , whe r e G ( x ) is c on- structe d by the p ar ameterize d inte ger re lation c onstruct ion algorithm as ab ove, 7 the u pp er b ound N on the height of its minimal p olynomial g ( x ) on an algebr aic numb er α ar e kn own. Then G ( α ) = 0 , (6) and the primitive p art of p olynomial G ( x ) is the minimal p olynomial of algebr aic numb er α . Pro of: P ro of is given by contradiction. Accor ding to L e mma 2, supp o se o n the contrary that G ( α ) 6 = 0 , then | G ( α ) | ≥ n − 1 · ( n + 1) − n + 1 2 · | G | − n ∞ · N 1 − n . F rom theor y 4, we get | G ( α ) | < n − 1 · ( n + 1) − n + 1 2 · | G | − n ∞ · N 1 − n . So, G ( α ) = 0. Since algebr aic num ber α is degree n > 0, then the pr imitive po lynomial of G ( x ) deno tes pp ( G ( x )), hence pp ( G ( x )) is just ir reducible and equal to g ( x ). O f co urse, it is unique. So, the c o rollar y is finished. 3.1 Reco v ering quadratic algebraic n um b er from appro x- imate v alue F or simplicity , we discus s how to o btain quadr a tic alge braic n um ber fr om its approximation using integer r elation a lgorithm. Let ˜ α be the a pproximate v alue, considering the vector v = (1 , ˜ α, ˜ α 2 ). O ur goal is to find a v ector w whic h has all in teger entries such that the do t pro duct of v a nd w is les s than a low er bo und, which is obtained a nd we are a ble to get the size of the neighbor ho o d is 1 / (12 √ 3 N 4 ) from theorem 4. The following theor em answers the ba sic ques tio ns of this appro ach. Theorem 5 L et ˜ α b e a n appr oximate value b elonging to an unknown quadr atic algebr aic numb er α , if ε = | α − ˜ α | < 1 / (1 2 √ 3 N 4 ) , (7) wher e N is the upp er b ound on the height of its minimal p olynomia l. Then G ( α ) = 0 , and the primitive p art of G ( x ) is its minimal p olynomial, wher e G ( x ) = P 2 i =0 a i x i is c onstructe d using inte ger r elation algorithm as ab ove. Pro of: F rom theorem 4 and corollary 2 , if and o nly if | G ( α ) | < n − 1 · ( n + 1) − n + 1 2 · | G | − n ∞ · N 1 − n , 8 Therefore, G ( α ) = 0. Under the assumption of the theorem w e get n = 2 and | G ( ˜ α ) | > 0, hence it is obvious that inequalit y (5)ho lds , i.e., 0 < n − 1 · ( n + 1) − n + 1 2 · | G | − n ∞ · N 1 − n − n · ε · | G | ∞ . (8) So, Solving inequality (8 ) yields ε < 1 / (1 2 √ 3 | G | 3 ∞ N ) . F rom theorem 3, and α is a quadr atic algebraic n umber, so | G | ∞ is just equal to N . So, the theo ry is finished. This theorem leads to the following algorithm for r e c ov er ing the qua dr atic algebraic num b er of ˜ α : Algorithm 2 Re c overing qu adr atic algebr aic nu mb er algori thm Input: an flo ating numb er ( ˜ α , N ) b elonging to an unknown quad r atic algebr aic numb er α , i .e., satisfying (7). Output: an quadr atic algebr aic n u mb er α . Step 1: Construct the ve ctor v ; Step 2: Compute ε satisfying (7); Step 3: Cal l algorithm 1 t o find an inte ger r elation w for v ; Step 4: Obtain w ( x ) the c orr esp onding p olynomial; Step 5: L et g ( x ) b e the primitive p art of w ( x ) ; Step 6: Solve the e quation g ( x ) = 0 and cho ose the c orr esp onding algebr aic numb er to α ; Step 7: r etur n α . Theorem 6 Algorithm 2 works c orr e ctly as sp e cifie d and uses O( l og N ) binary bit op er ations, wher e N is t he u pp er b ound of height on its minimal p olynomia l. Pro of: Correctness follows from theorem 5 . The cost o f the algo rithm is O( l og N ) binar y bit oper ations obviously . 9 3.2 Obtaining minimal p olynomial of high degree If α is a real num b er, then by definitioin α is algebraic exactly if, for so me n , the vector (1 , α, α 2 , · · · , α n ) (9) has an integer relatio n. The in teger coe fficient p olyno mial of low es t degree, whose ro ot α is, is determined uniquely up to a constant multiple; it is called the minim a l pol y nomi al for α . Int eger relation alg orithm can b e employed to search for minimal p olynomial in a straightforward way by simply feeding them the vector (9) as their input. Let ˜ α be a n approximate v alue b elonging to an unknown alg ebraic num b er α , considering the vector v = (1 , ˜ α, ˜ α 2 , · · · , ˜ α n ), how to obtain the exact minimal poly no mial from its approximate v alue? W e ha ve the same technique ans wer to the q uestion from the following theorem. Theorem 7 L et ˜ α b e an appr oximate value b elonging to an unknown algebr aic numb er α of de gr e e n > 0 , if ε = | α − ˜ α | < 1 / ( n 2 ( n + 1) n − 1 2 N 2 n ) , (10) wher e N is the upp er b ound on the height of its minimal p olynomia l. Then G ( α ) = 0 , and the primitive p art of G ( x ) is its minimal p olynomial, wher e G ( x ) = P n i =0 a i x i is c onstru cte d using the p ar ameterize d inte ger r elation c on- struction algo rithm as a b ove. Pro of: The proo f ca n b e given s imilarly to that in theor e m 5. It is e asiest to appr eciate the theorem by seeing how it justifies the following algorithm for obtaining minimal polynomia ls from its a pproximation: Algorithm 3 O bt aining minimal p olynomial algorithm Input: an flo ating nu mb er ( ˜ α , n , N ) b elong t o an unknown algebr aic n umb er α , i.e., s atisfying (10). Output: g ( x ) , the m inimal p olynomial of α . Step 1: Construct the ve ctor v ; Step 2: Compute ε satisfying (10); Step 3: Cal l algorithm 1 t o find an inte ger r elation w for v ; Step 4: Obtain w ( x ) the c orr esp onding p olynomial; Step 5: L et g ( x ) b e the primitive p art of w ( x ) ; Step 6: r etur n g ( x ) . 10 Theorem 8 Algorithm 3 works c orr e ctly as sp e cifie d and u ses O( n ( l og n + l og N ) ) binary bit op er ations, wher e n and N ar e the de gr e e and height of its minimal p olynomial, r esp e ct ively. Pro of: Correctness follows from theorem 7 . The cost o f the algo rithm is O( n ( l og n + l og N )) binary bit op er ations obviously . The metho d of obta ining minimal p olynomials from an appr oximate v alue ca n be extended to the set of complex num b er s and many applicatio ns in computer algebra and science. This yields a simple factorizatio n algo rithm for m ultiv ariate po lynomials with rational co efficients: W e can reduce a multiv ariate p olyno mial to a biv a riate po lynomial using the Hilber t irreducibility theorem, the basic idea was describ ed in [5 ], and then conv ert a biv ar iate p oly no mial to a univ a riate po lynomial b y substituting a transcendental num ber in [18] or an algebraic num b er of high degree for a v ariate in [8]. It c an find the biv ariate p olynomia l’s factor s, from which the factors of the or iginal multiv ar iate p olynomia l c an b e recovered using Hensel lifting. After this substitution we can get an approximate ro o t o f the univ ariate p olynomial and use our a lgorithm to find the ir reducible p o lynomial satisfied by the exact ro o t, which must then b e a factor of the g iven po lynomial. This is rep ea ted un til the factor s are found. The other yields a n efficient metho d o f conv erting the ratio nal approximation representation to the minimal p olynomial r epresentation. T he traditio nal repre- sentation o f a lgebraic num ber s is by their minimal p olyno mia ls [19, 20, 21, 22]. W e now prop ose an efficient metho d to the minimal p o lynomial repr esentation, which only needs an approximate v alue, deg ree and height of its minimal poly- nomial, i.e., a n orde r ed triple < ˜ α, n, N > instead of a n algebr aic num ber α , where ˜ α is its approximate v alue, and n and N are the deg ree and height o f its minimal p olynomial, resp ectively , denotes < α > = < ˜ α, n, N > . It is not ha rd to see the computations in the r epresentation ca n be c hang ed to computations in the o ther without lo s s of efficiency , the ra tional a pproximation metho d is clos e r to the intuitiv e notion of computation. 4 Exp erimen tal Results Our algorithms are implemented in Ma ple . The following e xamples run in the platform of Maple 1 1 and PIV3.0G,51 2M RA W. The firs t three ex amples illumi- nate how to obtain ex act quadra tic algebra ic num ber a nd minimal p olynomials. Example 4 tests o ur a lgorithm for factoring primitive p olynomia ls with integral co efficients. Example 1 Let α be an unknown quadra tic algebr a ic nu m be r . W e only 11 know an upp er b ound of heig ht on its minimal p o lynomial N = 47 . According to theorem 5, compute quadra tic alge braic num b er α a s follows: First obtain control err or ε = 1 / (1 2 ∗ √ 3 ∗ N 4 ) = 1 / (18077 2944 7692 ∗ √ 3) ≈ 1 . 0 × 10 − 8 . And then ass ume that we use some numerical metho d to get an approximation ˜ α = 1 1 . 9372 5393 3, s uch that | α − ˜ α | < ε . Calling algor ithm 2 yields as follows: Its minimal p olyno mia l is g ( x ) = x 2 − 8 ∗ x − 47. So, we can obtain the corres p o nding quadra tic a lgebraic num b er α = 4 + 3 √ 7. Remark 3 The function identif y in maple 11 ne e ds Digits =13, wher e as our algorithm only ne e ds 9 digits. Example 2 F or obtaining exa ct minimal p oly no mials from approximate ro ot ˜ α , we only know degree n = 3 and he ig ht N = 17 of its minimal p o lynomial. According to theor e m 7, just as do in Ex ample 1: First get the error ε = 1 / ( n 2 ( n + 1) n − 1 2 N 2 n ) = 1 / 69 5161 9872 ≈ 1 . 4 × 1 0 − 10 . Assume that we use some nu merical metho d to get an approximation ˜ α = 16 . 8080 3464 2702 , such that | α − ˜ α | < ε . Calling algorithm 3 yie lds a s follows: Its minimal p o ly nomial is g ( x ) = x 3 − 17 ∗ x 2 + 4 ∗ x − 13. Example 3 Let a known floating num b er ˜ α belo nging to some alge br aic nu m ber α of degre e n = 4, where ˜ α = 3 . 1462 6436 9941 98, we also know an upp er bo und of height N = 10 on its minimal p olyno mia l. According to theor em 7, we can get the err or ε = 1 / ( n 2 ( n + 1) n − 1 2 N 2 n ) = 1 / (4 2 ∗ 5 7 2 ∗ 10 8 ) ≈ 2 . 2 × 10 − 12 . Calling a lgorithm 3, if only the floating num ber ˜ α , such that | α − ˜ α | < ε , then we can g et its minimal poly nomial g ( x ) = x 4 − 10 ∗ x 2 + 1. So, the exa ct algebraic num b er α is able to denote < α > = < 3 . 146 2643 6994 1 98 , 4 , 1 0 > , i.e., < √ 2 + √ 3 > = < 3 . 1462 6436 9941 98 , 4 , 10 > . Example 4 This example is an application in factoring primitiv e p olyno mia ls ov er integral co efficients. F or the conv eniency of display in the pap er , we choose a very simple p oly no mial as follows: p = 3 x 9 − 9 x 8 + 3 x 7 + 6 x 5 − 27 x 4 + 21 x 3 + 30 x 2 − 21 x + 3 W e want to factor the polynomial p via reconstruction of minimal polynomia ls ov er the integers. First, w e tra nsform p to a primitiv e p olyno mial as follows: p = x 9 − 3 x 8 + x 7 + 2 x 5 − 9 x 4 + 7 x 3 + 10 x 2 − 7 x + 1 , W e s ee the upp er b o und of co efficients on p olynomial p is 10, which has re la tion with an upp er b ound of co efficients of the factors on the primitive polynomial p by Landa u-Mignotte b ound [23]. T aking N = 5, n = 2 yields ε = 1 / (2 2 ∗ (2 + 1) 2 − 1 2 ∗ 5 4 ) = 1 / (750 0 ∗ √ 3) ≈ 8 . 0 × 1 0 − 5 . Then w e compute the appro ximate ro ot on x . With Maple we get via [fsolve( p = 0 , x )]: S = [2 . 618 0339 89 , 1 . 250523220 , − . 9223475138 , . 3819660113 , . 2192 284350] 12 According to theorem 7, let ˜ α = 2 . 6 1 8033 989 be a n approximate v alue b elong ing to some quadra tic algebra ic num b er α , calling algorithm 3 yields as follows: p 1 = x 2 − 3 ∗ x + 1 . And then we use the p olyno mial division to get p 2 = x 7 + 2 ∗ x 3 − 3 ∗ x 2 − 4 ∗ x + 1 . Based on the Eisenstein’s Criterion [2 4], the p 2 is irreducible in Z [ X ]. So, the p 1 and p 2 are the factors o f pr imitive p olyno mial p . 5 Conclusion In this pap er, we hav e presented a new metho d for obtaining e x act results by nu merical approximate co mputations. The key technique of our metho d is bas ed on an improv ed para meterized int eger rela tion co nstruction algo rithm, which is able to find an exact rela tion by the a ccuracy control ε in for mula (10) is a n exp onential function in degr ee and heig ht of its minimal po lynomial. The re- sult is almost consis tent with the existence of error controlling on obta ining an exact rationa l num b er from its a pproximation in [1 5]. Using our algorithm, we hav e succeed in fac to ring mult iv ariate p o lynomials with ra tional co efficients a nd providing an efficient metho d of conv erting the rational approximation represen- tation to the minimal p o lynomial representation. 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