Factorization of Multivariate Polynomials by Extended Hensel Construction

Transcription

1 ACM SIGSAM Bulletn, Vol 9, No., March 5 Factorzaton of Multvarate Polynomals by Extended Hensel Constructon Daju Inaba [email protected] Doctoral Program n Mathematcs, Unversty of Tsukuba Tsukuba-sh, Ibarak 5-857, Japan Abstract The extended Hensel constructon s a Hensel constructon at an unlucky evaluaton pont for the generalzed Hensel constructon, and t allows as to avod shftng the orgn n multvarate polynomal factorzaton. We have mplemented a multvarate factorzaton algorthm whch s based on the extended Hensel constructon, by solvng a leadng coeffcent problem whch s pecular to our method. We descrbe the algorthm and present some expermental results. Experments show that the extended Hensel constructon s qute useful for factorng multvarate polynomals whch cause large expresson swell by shftng the orgn. Introducton Let K be a number feld and F(x,u,...,u l ) be a square-free polynomal over K. (In ths paper, K = Q for all the examples and experments.) The generalzed Hensel constructon breaks down f the expanson pont (s,...,s l ) K l s so chosen that F(x,s,...,s l ) s not square-free or the leadng coeffcent w.r.t. x, of F(x,u,...,u l ) vanshes at (s,...,s l ). In the frst case, such a pont (s,...,s l ) K l s called a sngular pont for the Hensel constructon, and n the latter case, t s sad that the leadng coeffcent s sngular at (s,...,s l ). One very mportant applcaton of Hensel constructon s the factorzaton of multvarate polynomals; see [Mus7] and [WR75]. We usually choose the orgn as the expanson pont of Hensel constructon, and f the orgn s a sngular pont or the leadng coeffcent s sngular at the orgn then we shft the orgn. Shftng the orgn s called the nonzero substtuton and t often causes a large expresson swell, makng the computaton very tme-consumng. It s called the nonzero substtuton problem, or the bad-zero problem n [GCL9]. For how serous the nonzero substtuton problem s, see polynomals n 6 below. The nonzero substtuton problem has been attacked by Wang [Wan77] and Kaltofen and Trager [KT9] et al., but t seems to the author that the problem has not been solved satsfactorly. In ths paper, we wll show that the extended Hensel constructon s an ultmate soluton to the nonzero substtuton problem. As for the Hensel constructon at a sngular pont, Kuo [Kuo89] descrbed a drect extenson of Hensel s lemma for bvarate polynomal F(x,u ), and Sasak and Kako [SK99] presented a method for F(x,u,...,u l ), l, so as to generalze the Puseux seres solutons of a bvarate algebrac equaton to the case of l. Sasak and Kako called ther method extended Hensel constructon (we abbrevate t to E.H.C. below) and the present author uses ths namng n ths paper. Sasak and Kako [SK99] nvestgated only monc polynomals (hence the leadng coeffcents are not sngular). Later, Sasak and Inaba [SI] extended Sasak-Kako s method so as to apply to the polynomals wth sngular leadng coeffcents. In the polynomal factorzaton, the E.H.C. s for both non-sngular and sngular leadng coeffcents are necessary. Sasak and Inaba [SI] proposed to utlze the E.H.C. for factorzaton of multvarate polynomals. However, no mplementaton has been made so far, and the present author has mplemented recently. It s well known that the so-called leadng coeffcent problem exsts n the polynomal factorzaton; see [GCL9]. In the conventonal methods, ths problem has been solved fully. In partcular, Wang [Wan77] solved ths problem as follows: factorze the leadng coeffcent, and assgn every factor of the leadng coeffcent to a Hensel factor so that the resultng Hensel factor becomes a factor of an rreducble polynomal. Wang s method necesstates us to choose an expanson pont sutably. It seems that Wang s method can be used n the case of E.H.C., but t s wrong, because we fx the expanson pont to the orgn. In ths paper, we solve the leadng coeffcent problem dfferently. 4

2 Inaba In, we survey the E.H.C. brefly. In, we outlne a multvarate factorzaton algorthm based on E.H.C.. In 4, we gve a soluton to the leadng coeffcent problem n the E.H.C.. In 5, we present several technques for speedng up the factorzaton based on E.H.C., and n 6, we test the new method by many examples. The reader wll see that the method s qute effcent for multvarate polynomals causng large expresson swell by the nonzero substtuton. Bref survey of the extended Hensel constructon Let x,u,...,u l be the varables, wth x the man varable and u,...,u l the sub-varables. Let K[u,...,u l ], K(u,...,u l ) and K{u,...,u l } be the rng of polynomals, the feld of ratonal functons and the rng of formal power seres, respectvely, over K, n varables u,...,u l. We abbrevate (u,...,u l ) and (s,...,s l ) to (u) and (s), respectvely. Let a gven polynomal F(x,u) K[x,u] be square-free, prmtve w.r.t. every varable, and expressed as F(x,u) = f n (u)x n + f n (u)x n + + f (u)x, f n (u). () By deg(f), lc(f) and tdeg( f ), we denote the degree, the leadng coeffcent of F w.r.t. the man varable x, and the total-degree of f w.r.t. u,...,u l, respectvely: f T = cu e ue l l, c K, then tdeg(t ) = e + + e l. By tdeg u (F) we denote the totaldegree of F w.r.t. u,...,u l. By ord( f ) we denote the order of f,.e., the mnmum of the total-degrees of terms of f. For the ratonal functon f (u)/g(u), we defne the order as ord( f /g) = ord( f ) ord(g). By gcd(f,g) we denote the greatest common dvsor of F and G. By cont(f) we denote the content of F(x,u) w.r.t. x,.e., cont(f) = gcd( f n, f n,, f ). By p, wth p a polynomal, we denote the deal generated by p. Let G(u) be a fnte or nfnte seres of ratonal functons such that G(u) = g (u)/d (u) + g (u)/d (u) + + g k (u)/d k (u) +, g k (u) and d k (u) are homogeneous polynomals n K[u], () ord(g k /d k ) = k (k =,,,...). By K{(u)} we denote the rng of seres of homogeneous ratonal functons of nonnegatve orders, such as G(u) n (). The reader can see examples of elements of Q{(u)}[x]. Defnton (sngular pont, sngular leadng coeffcent) We call (s) a sngular pont for the Hensel constructon, or a sngular pont n short, f F(x,s) s not square-free. If f n (s) = then we say that the leadng coeffcent s sngular at (s). In the E.H.C., t s convenent to ntroduce the total-degree varable t for sub-varables u,...,u l by the transformaton u tu ( =,...,l). (We may ntroduce the weghted total-degree varable t by the transformaton u t ω u ( =,...,l), where ω,...,ω l are postve ntegers). Next, we defne Newton lne and Newton polynomal for F(x,u) K[x,u] or F(x,u) K{u}[x], as follows; Defnton (Newton lne L New and Newton polynomal F New (x,u) for F(x,u)). For each monomal cx t j u j uj l l of F(x,tu), wth c K and j = j + + j l, plot a dot at the pont (, j) n the (e x,e t )-plane ;. Let L New be a straght lne n (e x,e t )-plane, such that t passes the pont (n,ν), wth ν = ord( f n ), and another dot plotted and that any dot plotted s not below L New ;. Construct F New (x,tu) by summng all the monomals whch are plotted on L New. Example The Newton lne and the Newton polynomal. F = x (y + z + z ) + x (y z + y ) + x (y z + z ) + y + z e t Fg. L New Fgure llustrates the Newton lne L New for the l.h.s. polynomal. Hence, the Newton polynomal for the l.h.s. polynomal s e x F New = x (y + z ). 4

3 Multvarate factorzaton by E.H.C. Let the slope of the Newton lne be λ. We put ˆn = f λ = else determne ntegers ˆν and ˆn (> ) to satsfy ˆν/ ˆn = λ and gcd( ˆn, ˆν ) =. Accordng to Lemma of [SK99], f we shft the Newton lne by step / ˆn n the e t -drecton successvely, all the nteger lattce ponts above the Newton lne are rdden by the shfted lnes. Therefore, we determne polynomal F(x, u, t) and deal Ī k as follows. F(x,u,t) def = F(x/tλ,tu) t ν nλ, () Ī k = t k/ ˆn, k =,,,. The transformaton moves the Newton lne on the e x -axs, and the Newton polynomal for F(x,u,t) s F New (x,u) = F(x,u,). (4) Next, we use a conventonal method to factorze F New (x,u) n K[x,u] as follows (we may well expert that, snce F New s restrcted very much, the non-zero substtuton to F New does not cause a large expresson swell): F New (x,u) = Ḡ () () (x,u) Ḡ r (x,u), r, Ḡ () = H m =,...,r, (5) gcd(ḡ (),Ḡ () j ) = for any j. Here, H s rreducble n K[x,u] and m s a postve nteger. Note that, f F New (x,u) s rreducble, then F(x,u) s also rreducble. Then, followng the procedure of E.H.C., we can construct Ḡ (k) (x,u,t) ( =,...,r), k =, such that F(x,u,t) Ḡ (k) (k) (x,u,t) Ḡ r (x,u,t) (mod Ī k+ ). (6) We descrbe the constructon procedure of Ḡ (k) (x, u, t) brefly; see [SK99] for detals. We frst calculate Moses-Yun s ( =,...,r; l =,,...,n) to satsfy polynomals W (l) W (l) F New Ḡ () + + W (l) r deg( W (l) ) < deg(ḡ () F New Ḡ () r = x l, ) ( =,...,r ). The reader can see several examples of Moses-Yun s polynomals n Example 4 n 5; for Moses-Yun s polynomals, see [MY7]. Suppose that we have constructed Ḡ (k ) (k =,,...,k ). Then, we calculate and construct Ḡ (k) = Ḡ (k ) + δḡ (k) ( =,...,r) as D (k) F Ḡ (k ) Ḡ (k ) r (mod Ī k+ ) = d n (k) x n + d (k) n xn + + d (k), (8) δḡ (k) = W (n) d n (k) For an actual performance of the E.H.C., see Example 4 n 5. Factorzaton based on the E.H.C. + W (n ) d (k) n + + W () d (k). (9) The E.H.C. allows us to perform the factorzaton of multvarate polynomals wthout the nonzero substtuton. In ths secton, we outlne our factorzaton method. Frst, we factorze the Newton polynomal F New n K[x,u] as follows. { FNew (x,u) = G (x,u) G (x,u) G r (x,u), () G = cont(f New )x n, gcd(g,g j ) = ( j). 44 (7)

4 Inaba Note that n s the degree w.r.t. x, of the lowest degree terms of F New : F New = ˆf n (u)x n + + ˆf n (u)x n, ˆf n (u). () In ths paper, we confne ourselves to the case that G (x,u),...,g r (x,u) are rreducble n K[x,u],.e. F New s square-free. Otherwse, t wll be dscussed n 7. Ths s a lmtaton of our current program, and removng ths lmtaton s a future theme. One may thnk that the factorzaton requres E.H.C. only once, but ths s wrong n general, as we wll see just bellow. Defnton (Newton polygon for G(x,u)) Let G(x,u) K[x,u] or G(x,u) K{(u)}[x]. For each term cx t j u j uj l l /D(tu) of G(x,tu), where c K, j = j + + j l and D(u) s a homogeneous polynomal n u,...,u l wth ord(d) = d, plot a dot at the pont (, j d) n the (e x,e t )-plane. The Newton polygon N for G(x,u) s a convex hull contanng all the dots plotted. Let the lne segments of N, facng the e x -axs, counted clockwse, be L,...,L ρ. Note that the Newton lne s nothng but L, the rghtmost lne segment facng the e x -axs. Fgure llustrates a Newton polygon havng two lne segments L and L facng the e x -axs. e t L L e x Fg. The Newton polygon for the polynomal F n Example 5. The Newton polygon may contan only one lne segment facng the e x -axs,.e. ρ =. In ths case, we perform the E.H.C. only once. Below, we dscuss the case of ρ >. Note that, n ths case, G (x,u) = cont(f New )x n, wth n. By applyng the E.H.C. to F(x,u) nfnte tmes wth ntal factors G j (x,u) ( j =,,...,r), F(x,u) can be factorzed as F(x,u) = G ( ) (x,u) G ( ) (x,u) G ( ) r (x,u). () Here, G ( ) (x,u) s a Hensel factor correspondng to cont(f New )x n, and G ( ) (x,u),...,g ( ) r (x, u) are rreducble elements n K{(u)}[x]; see Theorem (decomposton theorem) of [SI]. G ( ) may contan rreducble factors n K[x, u], or ts factors n K{(u)}[x] may be combned wth G ( ) ( r) to gve an rreducble factor n K[x,u]. Therefore, we must factorze G ( ) (x,u) further. Thus, we factorze the Newton polynomal for G ( ) (x, u) smlarly, and contnue ths procedure. Then, we obtan Hensel factors on the ρ lne segments successvely as L L L ρ. Next, we combne the Hensel factors constructed n K{(u)}[x]. We obtan an rreducble factor n K{u}[x] by the product of some rreducble factors n K{(u)}[x], then obtan an rreducble factor n K[x,u] by the product of some rreducble factors n K{u}[x]. Combnng elements of K{u}[x] to get an element of K[x,u] has been well nvestgated so far; the basc dea s to combne Hensel factors so that the sum of numercal coeffcents of same term becomes zero (see [SS9] or [Sas], for example). So, we descrbe how to combne Hensel factors n K{(u)}[x] to get an element of K{u}[x]. If the Hensel factor s n K{(u)}[x] then we call t ratonal, otherwse we call t ntegral. The followng strategy s adopted for combnng the ratonal Hensel factors.. Frst, for each {,...,ρ}, do the followng: f two or more Hensel factors on L have a denomnator factor d (u) whch s pecular to L (.e. not contaned n Hensel factors on L j ( j )), then combne Hensel factors contanng d (u) and elmnate t.. Next, f some Hensel factors on dfferent lne segments L,...,L m have the same denomnator factor d(u) then combne Hensel factors contanng d(u) and elmnate t. Fnally, removng the content of each factor computed n K[x,u], we obtan rreducble factors of F(x,u). We present examples of factorzaton n 4 and 5. 45

5 Multvarate factorzaton by E.H.C. 4 Leadng coeffcent problem n E.H.C. In ths secton, we present a method to solve the leadng coeffcent problem n the E.H.C. appled to the multvarate polynomal factorzaton. In addton to the assumptons on F(x,u), we assume further that F(x,u) s sngular at the orgn (u) = () and has a sngular leadng coeffcent there. Let the rreducble factorzaton of the leadng coeffcent f n (u), n K[u] be as follows. { fn (u) = w W (u) ν W s (u) ν s, () w K, gcd(w,w j ) = ( j). The E.H.C. s ambguous n that each factor W j may be assgned to any of the ntal factors G () ( =,...,r). In the factorzaton, however, we should assgn W,...,W s correctly to G ( =,...,r), as Wang does n [Wan77]. Unfortunately, Wang s method s not applcable to the E.H.C., and we must devse another method. Defnton 4 (prncpal terms) For a polynomal f (u), we defne prncpal terms pt( f ) to be the sum of all terms of order ord( f ). For the ratonal functon f (u)/g(u), we defne the prncpal terms as pt( f /g) = pt( f )/pt(g). Obvously, the prncpal terms satsfy the followng multplcaton rule. Example Prncpal terms. pt( f g) = pt( f ) pt(g). (4) g = u 4u u + 5u 6u + 5u : pt(g ) = 6u + 5u, g = u 6 u + : pt(g ) =, g = (6u 5 5u4 + 4u )/(u + u ) : pt(g ) = 4u /u. Note that the Newton polynomal s expressed as (see ()) Accordng to () and (5), we have F New (x,u) = pt( f n )x n + + pt( f n )x n. (5) lc(f New ) = lc(g ) lc(g r ) = w pt(w ) ν pt(w s ) ν s. We assgn the number w to G and each W j ( j s) to G ( r) f pt(w j ) dvdes lc(g ); f pt(w j ) µ dvdes lc(g ) then we assgn pt(w j ) µ to lc(g ). Note that W j may be assgned to several G s n general and the number of W j assgned may be greater than ν j. Accordngly, we modfy F and G,...,G r as n the conventonal treatment of the leadng coeffcent problem. That s, f W µ W µ s s s assgned to G then we multply pt(w µ W µ s s )/pt(g ) to G and replace the leadng coeffcent of the resultng polynomal by W µ W µ s s. Furthermore, we multply the product of all the over-assgned factors to F New and F. Example Solvng the leadng coeffcent problem. F = x (y + 8y z yz z + y z y z yz y z + 5y z + 6y z ) + x (4y yz 4z 6y 5y z 5yz z + 44y z 7y z + yz y z + 8y z + y z ) + x ( 5z + 6y 4yz 8z 8y + y z 4yz + 5z + 4y z 58y z + 8yz + y z 9y z 6y z ) + ( + y 6z 8y + 6yz + z + 4y z 6yz + 6z 8y z + 6y z + yz + y z 6y z y z ). L Fgure llustrates the Newton polygon for the l.h.s. polynomal, and the polygon has only one lne segment L facng the e x -axs. Hence, we have to perform the E.H.C. only once. e t Fg. e x 46

6 Inaba The Newton polynomal F New and ts rreducble factorzaton are as follows. F New = [x(y z) ] [x(y + z) + ] [x(y + z) + ] We put G = x(y z), G = x(y + z) + and G = x(y + z) +. The rreducble factorzaton of the leadng coeffcent lc(f) s as follows. lc(f) = ( y + z + yz)(y + z + yz)( y z + yz). We put W = y + z + yz, W = y + z + yz and W = y z + yz. Then pt(w ) = y + z, pt(w ) = y + z and pt(w ) = (y + z). W s assgned to only G because W lc(g ), W lc(g ) and W lc(g ). Snce pt(w ) = pt(w ), W and W are assgned to both G and G. Let C ( =,,) be the product of factors assgned to G. Then, we have C = W, C = W W, C = W W. Next, we multply the product of all the over-assgned factors to F. We put F as follows. We calculate G = pt(c )/pt(g ) G ( =,,) as follows. G = x(y z), F = (C C C )/lc(f) = W W F. G = x(y + z) (y + z), G = x(y + z) (y + z). We use F nstead of F n the E.H.C.. Performng the E.H.C. wth ntal factors G ( =,,) up to order, we obtan the followng results (we omt the total-degree varable t etc.). G () = ( y + z + yz) x + y + z + yz, G () = [6y z yz(y + z) (y + z) ] x + ( 4y z yz + z 4y z + 4yz 6y z ) = (yz + y + z)[ ( y z + yz) x + z yz], G () = [6y z yz(y + z) (y + z) ] x + ( y z + 6y + yz z 8y z + yz + y z ) = (yz y z)[ (y + z + yz) x + y + z + yz]. Removng the contents of G (), G() and G (), we obtan rreducble factorzaton of F(x,u) : F(x,y,z) = [ ( y + z + yz) x + y + z + yz] [ ( y z + yz) x + z yz] [ (y + z + yz) x + y + z + yz] 5 Several technques for effcent computaton 5. Combnng the ratonal Hensel factors to ntegral ones The E.H.C. of a multvarate polynomal usually generates ratonal Hensel factors, and the computaton of ratonal Hensel factors s tme-consumng. On the other hand, we want to obtan ntegral Hensel factors by combnng ratonal Hensel factors. If we combne ratonal Hensel factors to obtan ntegral Hensel factors, just when we encounter coeffcents of ratonal functons, the computaton wll be speeded up largely. Example 4 Combnng the ratonal Hensel factors effcently. F = x 4 + x (y z) + x ( 6yz z y z + yz ) + x ( 9y z 6y z + 4y z + yz ) + (8y z + y z 9y z 6y z ). e t L Fg. 4 e x 47

7 Multvarate factorzaton by E.H.C. Fgure 4 llustrates the Newton polygon for the l.h.s. polynomal, and the polygon has only one lne segment L facng the e x -axs. Hence, we have to perform the E.H.C. only once. The rreducble factorzaton of the Newton polynomal F New s F New = (x + y) (x z) (x + xz yz). We put G = x+y, G = x z and G = x +xz yz, and calculate Moses-Yun s polynomals W (l) ( =,,; l =,,,) as follows. = y(y + z)(y z), W () = z(y + z)(y z), W () W () W () = (y + z)(y z), W () = (y + z)(y z), W () y = (y + z)(y z), W () 4z = (y + z)(y z), W () W () = = = x (y z) + yz z yz(y z)(y z), x + y z (y z)(y z), x (y 4z) + yz (y z)(y z), W () 9y = (y + z)(y z), W () 8z = (y + z)(y z), W () = 4xz + 9yz yz (y z)(y z). As we see, the Moses-Yun s polynomals are polynomals n x wth coeffcents of ratonal functons n sub-varables, n general. Performng the E.H.C. wth ntal factors G ( =,,) up to order, we obtan the followng Hensel factors. G () = x + y yz y + z, G () = x z + yz y + z, G () = x + xz yz y z. We see that G () and G () contan y + z as the denomnator factor and the denomnators are the same. Hence, we combne G () and G (), obtanng G () G() x + x (y z) 6yz + yz. Then, we put G () 4 = x + x(y z) 6yz + yz. We see both G () and G () 4 dvde F. Therefore, they are rreducble factors n Q[x, y, z]. 5. Cuttng off the hgh total-degree terms n coeffcents We often need to perform the E.H.C. up to a consderably hgh degree, because the Newton lne s usually not horzontal. In ths and the next subsectons, we descrbe two deas to avod executon of the E.H.C. to a hgh degree. In the E.H.C., the Hensel lftng s made by ncreasng the power of t, the total-degree varable w.r.t. u,...,u l. However, t s also ncluded n the man varable x as n (). Hence, f λ n (), the E.H.C. generates hgher degree terms w.r.t. u,...,u l n coeffcents of ether hgher degree terms (when λ > ) or lower degree terms (when λ < ) w.r.t. x. In the factorzaton, however, such hgher degree terms w.r.t. u,...,u l are unnecessary. Therefore, we cutoff any term T f tdeg u (T ) > tdeg u (F)/. Note that the E.H.C. s possble even f we cutoff such hgh degree terms. Example 5 Cuttng off the hgh total-degree terms. F = x 4 + x ( y z + y z ) + x (yz + 4z + y 4 9y z + 6yz 9y z ) + x ( y z z y 5 + 5y 4 z z 5 + y 7 ) + yz + 4y 4 z. e t L Fg. 5 e x 48

8 Inaba Fgure 5 llustrates the Newton polygon for the l.h.s. polynomal, and the polygon has only one lne segment L facng the e x -axs. Hence, we have to perform the E.H.C. only once. Note that tdeg y,z (F) = 7. The rreducble factorzaton of the Newton polynomal F New s F New = (x xz + 5yz) (x xy + 4z ). We put G = x xz + 5yz and G = x xy + 4z. Performng the E.H.C. wth these ntal factors up to order, we obtan (we use the same notatons as above) δg () = xz + y 4, δg () = xy. However, we see deg(y 4 ) > 7/, hence we cut off y 4, obtanng G () = x + x ( z z ) + 5yz, G () = x + x ( y + y ) + 4z. We see that G () dvdes F and F/G () = x x (z+z )+5yz+y 4 also dvdes F. Therefore, G () and F/G () are rreducble factors n Q[x,y,z]. 5. E.H.C. from the left to the rght In, we descrbed the successve E.H.C. s on the lne segments L L L ρ. In each E.H.C., the Hensel lftng must be repeated untl each coeffcent w.r.t. x contans terms of a suffcently hgh total-degree. Thus, we must perform the E.H.C. to a consderably hgh degree w.r.t. the varable t, n partcular, when λ. We can avod the E.H.C. to a hgh degree f we combne the Hensel factors constructed from the rght sdes and those constructed from the left sdes (L ρ L ρ L ). In order to construct the Hensel factors from the left sdes, we apply the followng transformaton T Rev to F(x,u), perform the E.H.C. of the transformed polynomal, and apply the nverse transformaton TRev to the Hensel factors obtaned; see [SI]. T Rev : F(x,u) x deg(f) F(/x,u). (6) We explan the case of ρ =. Frst, we perform the E.H.C. from the rght to the left (L L ), then we obtan F(x,u) = G ( ) (x,u) G ( ) (x,u) G ( ) r (x,u). (7) Next, we perform the E.H.C. from the left to the rght (L L ). We apply the transformaton T Rev n (6) to F(x,u). Let F(x,u) = T Rev F. Note that the Newton lne for F(x,u) s nothng but L. We factorze the Newton polynomal F New n K[x,u] as follows. { F New (x,u) = H (x,u) H (x,u) H s (x,u), (8) H = cont( F New )x n n, gcd(h,h j ) = ( j). If s = then G ( ) (x,u) s rreducble and the factorzaton of F(x,u) s equvalent to the case of ρ =. Below, we dscuss the case of s. Applyng the E.H.C. to the Newton polynomal F New, we obtan For =,...,s, let H ( ) = T Rev F(x,u) = H ( ) (x,u) H ( ) (x,u) H ( ) s (x,u). (9) ( ) H. Accordng to Theorem n [SI], we have the correspondence {Hensel factors of G ( ) } { H ( ),..., H ( ) s }, wth ambguty of unts n K{(u)}. In the actual program, we remove the ambguty by normalzng the leadng coeffcent of H ( ) ( =,...,s) to lc(g ( ) ),.e., we multply lc(g ( ) )/lc( H ( ) ) to H ( ) (there wll be a better normalzaton method). Let the normalzed Hensel factor be H ( ) ( =,...,s). Fnally, we combne Hensel factors {G ( ),...,G ( ) r,h ( ),...,H s ( ) } to obtan rreducble factors of F(x,u). 49

9 Multvarate factorzaton by E.H.C. Example 6 Combng the Hensel factors constructed n both drectons. F = x 4 (y z ) + x (y + z + y + z ) + x ( + y 4z y + 5yz z + y + 6y z + z ) + x ( 5y 9y 5yz 5z + y + y z 5z ) + (y 5y 7y z yz y 4 y z yz z 4 ). () Fgure n llustrates a Newton polygon havng two lne segment L and L facng the e x -axs. We frst compute the Hensel factors on L. The rreducble factorzaton of the Newton polynomal F New s F New = x 4 (y z ) + x (y + z) x = x [x (y + z) ] [x (y z) + ]. We put G = x, G = x(y + z) and G = x(y z) +. Performng the E.H.C. wth these ntal factors up to order, we obtan the followng Hensel factors (we use the same notatons as above). G () = x + x (5y/ + y /4 5yz/ + 5z /) y /, G () = x (y + z) + y z y yz, G () = x (y z) + + y + z + y / yz/. Next, we perform the E.H.C. on L. We apply the transformaton T Rev n (6) to F, obtanng F and ts Newton polynomal F New as follows. F = x 4 (y 5y 7y z yz y 4 y z yz z 4 ) + x ( 5y 9y 5yz 5z + y + y z 5z ) + x ( + y 4z y + 5yz z + y + 6y z + z ) + x (y + z + y + z ) + (y z ), F New = x 4 y 5x y x = x (xy + ) (xy ). We put H = x, H = xy + and H = xy. We perform the E.H.C. wth these ntal factors up to order, and apply T Rev to the Hensel factors computed. Let the results be H () H () j = TRev () H ( =,): H () = x ( y + z + y + yz) + y + y yz + z, H () = x ( y z y / + yz/) + y y yz z. The leadng coeffcent of G () s, hence we remove the ambguty of unts by normalzng the leadng coeffcents ( j =,) to : we dvde H j by lc( H j ) (the power-seres dvson), obtanng H () H () /lc( H () ) x + y + 7y 4yz + z, H () H () /lc( H () ) x y/ + z/ + 5y /4 + yz/. Fnally, we obtan polynomal factors by combnng Hensel factors. In fact, calculatng the products G () obtan the followng rreducble polynomal factors of F. G () H() x (y + z) + x ( + y z) y y + yz z, H () ( =,), we G () H() x (y z) + x ( + y + z) y + y + yz + z. 6 Experments We have mplemented the followng three methods, one s our method and the others are conventonal methods, and tested the effcency of our method by comparng wth the other two methods. 5

10 Inaba method H Ths method employs the generalzed Hensel constructon. If the Hensel constructon breaks down at the orgn, the orgn s shfted. Except for one Hensel factor, the leadng coeffcents of other Hensel factors are converted to by the power-seres dvson. method W Ths method employs the generalzed Hensel constructon. The leadng coeffcent of F s factorzed and the factors are assgned correctly to the Hensel factors by Wang s method [Wan77]. method E Ths method employs the E.H.C., as descrbed above. In actual mplementaton, we must settle the cutoff degree of Hensel constructon. In methods H and W, we set the cutoff degree to tdeg u (F)/. In method E, we set the cutoff to degree tdeg u ( F)/ at the E.H.C. on the lne segment L ( ρ), where F s F [all extra factors of lc(f) that may appear n processng the leadng coeffcents of G,...,G r ]. We lst some features of these methods. method H It requres the unvarate factorzaton to get the ntal Hensel factors. It requres the nonzero substtuton at sngular ponts. If the leadng coeffcent of F s not a number, the Hensel factors become power seres always, because the nput polynomal s made monc by the power-seres dvson. method W It requres the unvarate factorzaton to get the ntal Hensel factors. It requres the nonzero substtuton at sngular ponts. It requres the multvarate factorzaton to factorze the leadng coeffcent. If the number of ntal Hensel factors s equal to the the number of rreducble factors of F(x,u), the Hensel factors are n K[x,u], and the computaton wll stop quckly. method E It requres the multvarate factorzaton of the Newton polynomal and the leadng coeffcent. It does not perform the nonzero substtuton. Ratonal functons wll often appear n coeffcents of the extended Hensel factors. If the Newton polygon contans ρ (> ) lne segments facng the e x -axs, t performs the E.H.C. ρ tmes. We have tested our algorthm on four dfferent knds of sample polynomals for whch the orgn s a sngular pont hence methods H and W perform the nonzero substtuton. st knd: The Newton polygon s such that ρ =, the Newton lne s of nonzero slope, and the leadng coeffcent s sngular. nd knd: The polynomal whch causes an extreme expresson swell by the nonzero substtuton. rd knd: The polynomal whch causes the case that [the number of Hensel factors] > [the number of rreducble polynomal factors]. 4th knd: The Newton polygon s such that ρ > and the leadng coeffcent s sngular. In all the experments, we use polynomals n three varables x, y and z. We do not use bvarate polynomals, because the expresson swell by the nonzero substtuton s not so large for bvarate polynomal. Furthermore, the Hensel factors of bvarate polynomals are always ntegral. In the samples of st and nd knds, [the number of Hensel factors] = [the number of rreducble polynomal factors]. Hence, these samples are rather specal and they are chosen for testng the performance of our programs. The samples of rd and 4th knds are rather general. We explan each experment and the correspondng result. We measured the computaton tme by averagng ten executons. The computatonal envronment s as follows. 5

11 Multvarate factorzaton by E.H.C. OS Lnux.4. CPU AMD Athlon(tm) XP 9+ (.6GHz) Memory. Gbyte Lbrary GAL(General Algebrac Language) ( GAL s an algebra system constructed by Sasak and Kako.) Experment On polynomals of the st knd. We generated polynomals F,...,F as follows. Each sample s the product of two rreducble polynomals contanng 5 terms, of total-degrees or less w.r.t. sub-varables y and z, and the coeffcents are chosen randomly from {, 9,...,9,}. For example, F = [ x 4 (9y z 4 6y z ) + x ( 8y 4 z 7 9z 4 ) + x (y z 6 + 4yz) + x (9y z 4 + 8yz 7 ) + yz ] [ x y z + x ( 8y z 9 + 7yz ) + x (y z 5 8y ) + 8yz z ]. Table I shows the result of experment, where T H, T W and T E are the average CPU tmes by methods H, W and E, respectvely. The rght columns show the ratos T H /T E and T W /T E. T H (sec) T W (sec) T E (sec) T H /T E T W /T E F F F F F F F F F F Table I : The average CPU tmes for F F. The nonzero substtuton converts each sample to a 8 tmes larger polynomal. We see that our algorthm shows a good effcency on these samples, and the largest reason for ths s that our method avods the nonzero substtuton. It should be ponted out that the number of Hensel factors are for each sample, hence these samples are very favorable to methods W and E. Experment We used the followng polynomal n the experment. P k = [ x y z + x (y k + z k ) + y + z z y k/ z k/ ] [ x y z + x (y k + z k ) y 5z + 4y + y k/ z k/ ]. As k becomes a larger, P k causes larger expresson smell by the nonzero substtuton. In the experment, we ncreased k as k = 5. The results are shown n Table II. k T H (sec) T W (sec) T E (sec) T H /T E T W /T E Table II: The average CPU tmes for P P 5. The method E s not so effcent when k s small. For larger k, the nonzero substtuton causes a large expresson swell, and the method E shows a very nce effcency. For k = 5, the number of terms of the shfted polynomal explodes to about 5 tmes by the nonzero substtuton. 5

12 Inaba Experment We used the followng polynomal n the experment. Q k = [ (x ( y + z ) + 5yz)(x (y + 4z) + ) + (x 7) (y k z k y k z k ) ] [ (x (y + 4z ) + yz)(x (y + z) + 7) (x + 5) (y k z k y k z k ) ]. In methods H and W, we set the expanson pont as (y,z) = (,). In three methods H, W and E, the number of Hensel factors of Q k s 4. In the experment, we ncreased k as k = The results are shown n Table III. k T H (sec) T W (sec) T E (sec) T H /T E T W /T E Table III: The average CPU tmes for Q 5 Q 5. We see that the method E shows a very nce effcency. The nonzero substtuton converts Q k to a 5 4 tmes larger polynomal. In the case of [the number of Hensel factors] > [the number of rreducble polynomal factors], both methods H and W become more and more tme-consumng as k ncreases. In method E, however, ratonal functons appear n the above samples, and the computaton s speeded up largely by the technque descrbed n 5.. Experment 4 On polynomals of the 4th knd. We generated polynomals H,...,H as follows. Each sample s the product of four rreducble polynomals contanng 8 terms, of total-degrees 4 or less w.r.t. sub-varables y and z, and the coeffcents are chosen randomly from { 4,,...,,4}. For example, H = [ x ( + yz) xy 5 z 4 + y z y 5 z 6 ] [ x yz + x ( y z + y 5 z 4 ) + 4yz ] [ x ( + yz) + y + z] [ x (y z) ]. T H (sec) T W (sec) T E (sec) T H /T E T W /T E H H H H H H H H H H Table IV: The average CPU tmes for H H. The nonzero substtuton converts each sample to a tmes larger polynomal. In method E, the E.H.C. s performed twce, and n each E.H.C. the cutoff order s set to tdeg u (H )/. We see that the method E stll shows a good effcency. Ths means that the technques presented n 5. and 5. are effectve for speedng up the method E. 5

13 Multvarate factorzaton by E.H.C. 7 Dscussons Tmng data n Table II show clearly that our method s qute effcent for polynomals causng large expresson swell by the nonzero substtuton. The data n Table III show clearly that our method s qute effcent n the case that [the number of Hensel factors] > [the number of rreducble polynomal factors]. And the data n Table IV show that our method s also effcent for polynomals for whch the Newton polygons are such that ρ =. We have not fully tuned up our program. In partcular, we have not tested the followng deas.. By changng the weghts for sub-varables, we can construct several Newton polynomals for one nput polynomal. Even f some Newton polynomal has many factors, another Newton polynomal may have less factors. Furthermore, we may be able to connect factors of dfferent Newton polynomals.. Our method descrbed above fals to gve rreducble factorzaton f the Newton polynomal s not square-free. In [SK99], a method s descrbed to treat the case of non square-free Newton polynomals, but the method s qute complcated because t ntroduces algebrac functons. Iwam [Iwa, Iwa4] nvestgated recently the E.H.C. for the case that the Newton polynomal has duplcated factors, and we may employ Iwam s method. There s a good possblty that our method wll be speeded up further. Acknowledgment The author thanks very much Prof. Tateak Sasak for supervsng ths work and varous advces. References [GCL9] K. O. Geddes, S. R. Czapor and G. Labahn: Algorthms for computer algebra. Kluwer Academc Publshers, 99. [Iwa] M. Iwam: Analytc factorzaton of the multvarate polynomal. Proc. CASC (Computer Algebra n Scentfc Computng), Eds. V. G. Ganzha, E. W. Mayr and E. V. Vorozhtsov, -5 (). [Iwa4] M. Iwam: Extenson of expanson base algorthm to multvarate analytc factorzaton. Proc. CASC 4 (Computer Algebra n Scentfc Computng), Eds. V. G. Ganzha, E. W. Mayr and E. V. Vorozhtsov, 69-8 (4). [KT9] E. Kaltofen and B. M. Trager: Computng wth polynomals gven by black boxes for ther evaluatons: greatest common dvsors, factorzaton, separaton of numerators and denomnators. J. Symb. Comput. 9, - (99). [Kuo89] T.-C. Kuo: Generalzed Newton-Puseux theory and Hensel s lemma n C[[x, y]]. Canad. J. Math. Vol. XLI, -6 (989). [Mus7] D. R. Musser: Algorthms for polynomal factorzatons. Ph. D. Thess, Unversty of Wsconsn, 97. [MY7] J. Moses and D. Y. Y. Yun: The EZGCD algorthm. Proc. 97 ACM Natonal Conference, ACM, (97). [Sas] T. Sasak: Approxmate multvarate polynomal factorzaton based on zero-sum relatons. Proc. ISSAC (Internatonal Symposum on Symbolc and Algebrac Computaton), ACM, 84-9 (). [SI] T. Sasak and D. Inaba: Hensel constructon of F(x,u,...,u l ),l, at a sngular pont and ts applcatons. ACM SIGSAM Bulletn 4, 9-7 (). [SK99] T. Sasak and F. Kako: Solvng multvarate algebrac equaton by Hensel constructon. Japan J. Indus. Appl. Math. 6, (999). [SS9] T. Sasak and M. Sasak: A unfed method for multvarate polynomal factorzatons. Japan J. Indus. Appl. Math., -9 (99). [Wan77] P. S. Wang: Preservng sparseness n multvarate polynomal factorzaton. Proc. 977 MACSYMA Users Conference, MIT, 55-6 (977). [WR75] P. S. Wang and L. P. Rothschld, Factorng multvarate polynomals over the ntegers. Math. Comp. 9, (975). 54