analysis of approximate factorization algorithm i

Japan J. Indu5t. Appl. Math., 9 (1992), 351-368

Analysis of Approx]mAte Factorization Algorithm I*

T a t e a k i SASAKI ~, Tomokatsu SAITO ~~ a n d Teruhiko HILANO ~~~

t lnstitute of Mathernatics, University of Tsukuba, Tsu~ba-8hi, Ibaraki 305, Japan

t~ Departrnent of Mathernatics, Sophia University, Kioi-cho, Chiyoda-ku, TokŸ 102, Japan

~~~ Kanagawa Institute of Technology, Atsugi-shi, Kanagawa 2~3-01, Japan

Received August 7, 1991 Revised December 20, 1991

In [2], a concept of approximate factorization of multivariate polynomial was introduced and two algo¡ of approximate facto¡ were proposed. One algorithm determines the irreducible factors by handling the combinations of roots of the forro A 1~~ +. �9 -+ An~ i , where ~1,...,qo,~ are the roots of a given polynomial, A l , . . . , A h are numbers, and i = 1, 2 , . . . , and it seems to be practical and important. However, [2] gave only an introductory doscription of the algorithm and the mathematical as well a~ computational analysis of the algorithm was postponed. This paper proves completen~ss of the algorithm by assuming ttmt the numerical coefficients are calculated with ah enough accuracy.

Key word~, approximate algebra, approximate factorization, computer algebra, polynomial factorization

1. I n t r o d u c t i o n

In a previous paper, [2], one of the authors (T.S.) and his coUaborators introduced a concept of approximate factorization of multivariate polynomials. The approximate factorization is an extension of conventional polynomial factorization, and it can be applied to polynomials with not only exact but also approximate coefficients such as represented in floating-point numbers. It should be emphasized that the approximate factorization is a wider concept than factorization of polynomials with approximate coetficients. In the latter factorization, the errors in numeric coefl3cients determine the "error term" and the error term is usually very small; in the approximate factorization, on the other hand, the error term may be determined independently of the accuracy of numeric coefficients, hence it may be as large as 0(10 -3) or even 0(10 -2) compared with the '~main t e rm ' . Furthermore, the approximate factorization leads us to a new approach to the exact factorization.

The authors of [2] proposed two algorithms of approximate factorization. One is elementary but time-cons,ming: it constructs a combinatorial number of candi- dates of approximate factors and performs trial division, hence its time-complexity is of exponential order w.r.t, the degree of input polynomial. Another algorithm determines the approximately irreducible factors by calculating approximately dependent linear combinations of numeric vectors, hence it seems to be efficient even

* Work supported in part by Japanese Ministry of Education, Science and Culture under grants #0355s00s.

352 T. SASAKI, T. SAITO and T. HILANO

for high degree polynomials. In this paper, by approx imate f a c t o r i z a t w n algori thm,

we m e a n only the lat ter a lgor i thm m e n t i o n e d above.

In [2], the authors gave only aja introductory desc¡ of the approximate factorization algorithm, and mathemat ical as well as computat ional analysis was postponed to coming papers. In this paper, we give a mathemat ica l analysis of the algorithm, by a s s u m i n g that the n u m b e r s appearing in the express ions ate calculated

wi th ah enough accuracy. Furthermore, in most of this paper (in Sections 2 ~., 5), we consider ouly the exact factorization. This does not mean tha t we analyze the approximate … only slightly but that our analysis is mostly common to both exact and approximate factorizations.

In 2, we explain the approximate factorization algorithm briefly and point out four essential problems which are on homogeneous relations among the roots of a given multivariate polynomial. In 3, we analyze the relations among roots and solve the first problem presented in 2. In 4, we derive a useful degree bound for factor polynomials. Using the degree bound, in 5, we analyze the relations among the roots which ate expanded into infinite power series in sub-variables and "approximated" by discarding higher terms. Finally, in 6, we prove completeness of the approximate factorization algorithm given in [2] by using the analysis of exact factorization.

2. P r e v i o u s W o r k and P r o b l e m s

By ( u , . . . , v) we denote the polynomial ideal generated by variables u , . . . , v. By C [ u , . . . , v] and by C { u , . . . , v} we denote the polynomial ring and power se¡ ring, respectively, and by C{u, . . . ,v}("~) wc denote the power series ring with terms in ( u , . . . , v) m+l discarded, over the complex numbers C. For monomial T -- u e . . . . v ~- , the total-degree w.r.t, u , . . . , v of T is the exponent sum e~ -{- �9 �9 �9 + e~ : tdeg(T) = e~ + . . - + e. . The total-degree of the polynomial is the maximum of total-degrees of its terms.

Let F ( x , u , . . . , v ) be a multivariate polynomial in C [ x , u , . . . ,v], with main variable x, and let the degree of F w.r.t, x be n : deg(F) = n. F is called m o n i c ir

its leading coefficient w.r.t, x is 1, tha t is

F ( :~ ,u , . . . ,v) = f , , ( ,~ , . . . , v )x" + f , ,_ , ( ,~ , . . . , v ) x " - ~ +

- - - + f0(~, .. ,v), (2.1)

A ( ~ , . . . , v ) = 1.

F is calle(] square-free i f i t has no multiple factor, and ifso then F ( x , Uo, . . . , vo)

is square-free for almost al] numbers u 0 , . . . , v0 in C. As was proved in [2], we can assume without loss of generality tha t F is monic and F ( x , 0 , . . . , 0) is square-free.

By m m c (F) we denote the absolute value of the m a x i m u m m a g n i t u d e n u m e r -

ical coej~icient of F. Let c be a small positive number, 0 < e �88 1. Ir F can be expressed as

F ( x , u, . . . , v) = G ( x , u , . . . , v ) H ( x , u, . . . , v) + S F ( x , u, . . . , v) , (2.2)

mmc ( 5 F ) / m m c (F) = O(s),

Analysis of Approximate Factorization 353

where G, H and 5F are elements of C[x, u , . . . , vi, then we say that F is factorized approximately into GH with accuracy O(e).

Let the roots o f F w.r.t, x be ~1, . . . ,~ ,~ , where Vi r ~j for any i r j by assumption. The ~1 , . . . ,Un can be expressed as elements of C { u , . . . ,v}, and we can actually coustruct ~ Ÿ ~~ in C { u , . . . , v} (m) such that

F ( x , u , . . . , v ) =- [ x - ~ Ÿ [ x - ~ ~ ( u , . . . , v ) ]

(mod ( u , . . . , v )m+ l ) . (2.3)

For a coustruction method, see [2]. The approximate factorization algorithm proposed by [2] utilizes { ~ Ÿ ~~}

satisfying (2.3), but the essence of it is to find the combinatious of roots satisfying

{ AI~I + "'" -b An~n = hi,

A1~1 ~ + +A.~~ = h~,

(2.4)

where Ai E C, i = 1 , . . . , n , and hi E C [ u , . . . , v ] , j = 1 , . . . , k . Let us explain (2.4) briefly. Let v i , . . . , vr, Vr+l be integers such that

1 =/21 < /22 < " '" < 12 r < Z/r+ 1 - - - n + l . (2.5)

Without loss of generality, we assume that F is factorized into r irreducible factors over C a s

F=F1...F~,

F i : ( x - - ~ v , ) ( x - - ~ v , + l ) " " " (x -- (flv,+l-1), i - - - -1 , . . . , r . (2.6)

Then, regarding A, and h~ to be ,mknowns in (2.4), we see that (2.4) has solutions

Av, = Av,+I . . . . . A~I+I-- 1 = A (i), i = 1 , . . . , r ,

h i = A(1) (~t::~il -~-...-~- ~2Jv~_l) Av.. .-{- A(r)((~i" -[-. . .-~- (~i .+1-1), (2.7)

j = l , . . . , k .

In many cases, system (2.4) has no solution other than (2.7) even for k -- 1. In this case, al] the irreducible polynomial factors of F can be determined by solving (2.4). However, we can construct e• in which (2.4) with k = 1 has solutions other than (2.7), see [2] for examples. Therefore, the essential problems for the algorithm given in [2] are as follows:

1. For some integer k and for any multivariate polynomial F, do the combinations of roots satisfying (2.4) determine al] the irreducible polynomial factors of F tmiquely?

2. Ir so, is it possible to construct all the irreducible polynomial factors of F by handling "approximate" roots ~ Ÿ ~~ in C { u , . . . , v} (L) for some positive integer L?

354 T. SASAKI, T. SAITO a n d T. HILANO

3. lŸ the above s ta tements 1 and 2 are true, then which value of k is large enough to factorize a given polynomial F ?

4. How much accuracy of the numeric coefficients is necessary to perform the approximate factorizat ion with a given accuracy?

In the following sections, we solve problems 1 and 2 positively; the answers to problems 3 and 4 are pos tponed to coming papers.

3. Combinat ions of R o o t s Giving Factors

In this section, we prove tha t if we set k -- n in (2.4) then combinat ions of roots satisfying (2.4) determine al] the irreducible polynomial factors of F uniquely.

Below, we wTite F(x , u , . . . , v) as F(x) . Let ~ 1 , - . . , ~n bc the roots w.r.t, x of polynomial F(x) which is defined as in 2, and let g l , . . . , gn be

gi = ~~ + . . . + ~ ~ , i = 1 , . . . , n . (3.1)

Note tha t gi

Z l ~ . . . ~ Z n :

E C [ u , . . . , v ] . Consider the sys tem of equations w.r.t, unknowns

{ G ([) = Zl + . . - ~- Zn - g l - - 0,

n G (n) : z I ~- ' ' ' -~- z n - g,+ = O.

(3.2)

LEMMA 3.1. System (3.2) has n! sotutions which ate

{(Z 1 = ~ 0 , 1 , . . . ,ZŸ 1 = ~ i ~ ) I { / 1 , . . . , i?l } = { 1 , . . . , n } } . (3.3)

Pro@ System (3.2) is equivalent to the sys tem (see Appendix A)

{ Z l -~- Z2 " [ - ' ' " ~- Zn = - - f n - - 1 ,

Z l Z 2 "~ " ' " ~- Z l Z n -~- " ' " "~ Z n - - l Z n = f n - - 2 ,

z lz2""z ,~ = ( - 1 ) " f 0 ,

where fi , i = n - 1 , . . . , 0, are defined in (2.1). This system gives

(Z -- Z l ) " "" (Z - - Z n ) mm Z n "[-- f n - - 1 z n - 1 ~ - - ' ' " "4-- l o = F ( Z ) .

(3.4)

For any solution (Zl = Z l , - . - , Zn = ~ n ) of (3.4), the left hand side of this equat ion becomes 0 at z = ~i, i = 1 , . . . ,n , and the right hand side becomes 0 only at z ---- ~i, i = 1 , . . . , n . Hence, we have the lemma. �9

In addit ion to (3.2), we consider the sys tem

{ H 0) = AlZl + ' " + A~zn - hi = 0,

. . . (3.5)

H (k) = )XlZkl - - ~ . . . --~ ~nZkn - - h k = 0,


and combined system

{ G (1) . . . . . G ( ') = 0, (3.6)

H (1) . . . . . H (k) = 0,

where k is a positive integer, hi E C, i -- 1 , . . . , n , and hi C C [ u , . . . , v ] , j = 1 , . . . , k. The hi and hi are t reated a s a constant a n d a given polynomial, respectively, in (3.5). Suppose we have a relation (2.4) among the roots ~Ol,... , ~on. Then, the combined system (3.6) has a solution (za = ~Ol,..., z,~ -- ~,~). Conversely, any solution of (3.6) satisfies a relation of the form (2.4) among the roots. Hence, we can investigate all the relations of the form (2.4) by investigating the solutions of

(3.6). The eombined system (3.6) has solutions for suitably ehosen hi and hi, as

expressions in (2.7) show. The important point is that the combined system may have solutions even ir h i a n d hi are chosen differently from those in (2.7).

THEOREM 3.2. Let k = n for the sys tem (2.4). Then, (2.4) holds iff, among the elements of { h i , . . . , h,~}, all the elements that correspond to the roots o f each irreducible polynomial factor, over C [ u , . . . , v], of F ate the same. Here, %i corresponds to a root qoj" means that their Ÿ ate the same, i.e., i = j . ( In other words, ir the factorization over C[ u , . . . ,v] /s given by (2.6) then the sys tem (2.4) has solutions for only sueh hj and hi, j = 1 , . . . , n, that ate given by (2.7).)

Pro@ Without loas of generality, we assume that F is factorized hato irreducible factors over C[u, . . . , v] as in (2.6).

We consider combined system (3.6). System (3.2) defines an algebraic variety V which consists of n! distinct points given by (3.3), and V can be decomposed into irreducible varieties uniquely; see [3], Sec. 13. Then, the system (3.5) selects a subset of V iŸ the combined system has solutions. The smallest subset tha t is an irreducible variety is such that if (Zl = ~Ol,..., z , = ~o~) is a solution of (3.6) then, for each i E { 1 , . . . , r } , some symmetr ic permutat ions among the elements of {qo.,, ~o.,+1,. . . , ~o.~+1-1} also give solutions of (3.6). In fact, using the method described in Appendix B, we can convert system (3.2) into

{F(Zl ) = O, G2(z2,zl) ~-- 0,... , G n ( z n , z n - 1 , . . . ,Zl) = O}, (3.7)

where G 2 , . . . , G,~ are given in Appendix B, hence if Zl is so chosen tha t Fl(Zl) = 0 then zl is either qo1,~o2,..., or ~o~-1 and the set {qo1,~o2,...,~ov2-1} cannot be divided any more because F1 is irreducible.

On the other hand, we can rewrite (2.4) with k = n as

/12 )(i)(i 1) (3.s)


This n • n matrix is a Vandermonde's matrix, and it is regular because ~Jl :~ ~j2 for any j l r J2. This means that the n-tuple (Al, A2, . . . , A,) satisfying (3.8) is unique for given ~j and hi, j = 1 , . . . , n.

Now, if the system (2.4) holds then we have other systems which are the same as (2.4) except that, for each i 6 {1 , . . . , r}, the elements of {~~,, ~ . , + 1 , . . - , ~~~+1-1} are symmetrically permuted in a suitable way, as we have noted above. This per- mutation is equivalent to permute the elements of {A,,, Au,+l , . . . , A~,+1-1} in (3.8) in the same way. Therefore, by the uniqueness of the solution of linear system (3.8), we have A., = Av~+l . . . . . )~ui+l-1 for each i 6 { 1 , . . . , r}. �9

COROLLARY 3.3. Let F be facto"ed over C a s in (2.6) with (2.5). Then, polynomials hi, i = 1 , . . . , n, satisfying (2.4) u¡ k = n a.e only those given in (2.7).

Theorem 3.2 shows that , if we find combinations of roots satisfying (2.4) and separate the combinations into "mutually disjoint combinations' , then we can obtain the combinations of roots corresponding to a l / the irreducible factors of F (for the defmition a n d a construction method of "mutually disjoint combinations' , see [2]). Therefore, Theorem 3.2 answers to the problem 1 in 2.

4. D e g r e e B o u n d for C o e f f i c i e n t s

Before going to solve the problem 2, we derive a useflfl degree bound for coefficients of factor polynomials of F. This degree bound plays an essential role in the approximate factorization algorithm.

DEFINITION 4.1. Let F be a polynomial given in (2.1), and let (eo, e l , . . . , en) be ah (n + 1)-tuple of non-negative numbers. We say that F is coeff-bounded by (eo, e l , . . . , e , ) w.r.t. ~ iŸ

tdeg(f, ,_i) <__ ei, i = O, 1 , . . . , n . (4.1)

From now on, we put

e, = t d e g ( f , _ , ) , / = 0, 1 , . . . , n . (4.2)

Note that e0 -- 0 since F is monic.

DEFIN1TION 4.2. We define numbers E 1 , . . . , E , as follows.

[ El = el,

= max{e2, 2el}, (4.3)

En ---- max {en, e ,_ l + el, en-2 + e2, en-2 + 2e l , . . . , nel }.

He.e, elements of the set for Ei range over all the partitions of integer i.

DEFINITION 4.3. Let E 1 , . . . , E , be defined az in (4.3), and let

c = max{E1/1, E 2 / 2 , . . . , En~n}. (4.4)

Analysis of Approximate Facto¡

W e d e f i n e n u m b e r s E l , . . . , E , as f o l l ( n v s .

E i = c • i - - 1 , 2 , . . . , n .

By definition, we have

357

(4.5)

e~<:E i_<Ei , i - - 1 , 2 , . . . , n . (4.6)

It is convenient to plot the pair (i, E~), or (i, Ei) in a two-dimensional coordinate system: the horizontal axis shows the index i and the vertical axis shows the total-degree of the corresponding coefficient. We call this coordinate system (i,tdeg)-plane. Figure 4.1 illustrates the relationship between two tuples (0, El , E 2 , . . . , E,,) and (0, El , E2 , - . . , En) in the (i, tdeg)-plane.

tdeg(coeff)

i Fig. 4.1.

Let A and B be the foUowing polynomials

A = x m + a ~ _ l x m - l + . . . + ao , r o < n ,

B = x ' ~ - m + b , _ , ~ _ l X " - m - 1 + �9 . . + bo, (4.7)

where a~, b~ E C [ u , . . . , vi, i = 0, 1 , . . . . In the following, we define aj = bj = 0 for j < 0 .

LEMMA 4.1. L e t F = A B . I r

tdeg(am_i + b n _ m _ i ) = max{tdeg(a~_,) , tdeg(b . . . . ,)},

i - - 1 , 2 , . . . , n , (4.8)


then A and B ate coeff-bounded by (0, E l , E s , . . . , En) w.r.t, x.

Proof. The coefficients of F = A B w.r.t , x are

fn--1 mm aro--1 ~- b n - m - 1 ,

f ,~-s = aro-2 + b . . . . 2 + a m - l b . . . . 1,

The first equat ion and the condit ion (4.8) give

m a x { t d e g ( a m _ l ) , t d e g ( b , - m - 1 ) } -- t d e g ( f ~ _ l ) -- e I ---- E 1.

Then, the above second equat ion and (4.8) give

max{ tdeg (am_2) , t d e g ( b n - m - s ) }

< max{ tdeg ( f~_2) , t d e g ( a m _ l ) + t d e g ( b , _ m _ l ) }

_< max{es, 2el} = Es.

Similarly, for i -- 3 , . . . , n, we obta in

max{ tdeg (am_ , ) , tdeg(bn_m_~)} < E,.

Condi t ion (4.8) means t ha t some leading t e rms of am-~ and b,~-m-~ remain uncanceled in A B , and this does not always hold a l though it holds in m a n y ac tual cases. In order to see how A and B are coeff-bounded in a general case, we mus t consider the cancel la t ion of leading t e rms of coefficients.

THEOREM 4.2. Let E 1 , . . - , E ~ be defined as in Del. 4.2, and let F = A B ,

where F is monic and given by (2.1) and A, B �9 C [ u , . . . ,v][x]. Then, A and B ate

co~Z-bo~~d~d ~ (0, El, E~,..., E,) ~ . ~ . t . x .

Proof. If we have equalities (4.8) for i = 1 , . . . , n, then the t heo rem is valid by L e m m a 4.1 because E~ <: E~, i -- 1 , . . . , n. So, we have only to consider the case t ha t the leading t e rms of coefficients of x ~ t e r m in A B cancel one another .

I. Let ~ be a posi t ive integer and t~ be a s t ra ight line, in the (i, tdeg)-plane, which connects the origin and the point (~, tdeg(a,~_~)) . If

1. for each i -- 1 , . . . , ~, point (i, tdeg(f,~_~)) is below ~~, and

2. for each i = 1 , . . . , ~ - 1, points (i, t deg (am_, ) ) and (i, tdeg(bn_m_, ) ) are below g~,

then t d e g ( a ~ _ ~ ) = t deg (b ,_m_~) , and the leading t e rms of a m - ~ and b , - m - ~ cancel each other. In order to see this, consider the coefficients of x n -~ t e rms of F - - A B :

f ,~_~ = a,~_,~ -~- b,~_,,~_~ + a,~- l b n - m - ~ + l -~- "'" + a,,~-,~ + l b,~-,~- l.


The above conditions 1 and 2 mean that

tdeg(fi~_~) < tdeg(am_~),

tdeg(a~_~b . . . . . +~) < tdeg(am_~), i = 1 . . . . , n - 1.

These inequalities lead us to the above c]aim directly.

II. Let ~ be a straight line which passes the points (i, E~), i -- 1 , . . . , n. Let D = {(i, tdeg(am_,)) , (i, t d e g ( b n - m - i ) ) l i = 1 , . . . ,n}, and let ~ be a line passing the origin of (i, tdeg)-plane and satisfying the following conditions.

1. ~ passes at least one of the points in D, 2. no point in D is above ~.

Now, suppose that the line ~ is above ~ in the region i > 0 of (i, tdeg)-plane, contracting the claim of theorem. Let n be a positive integer s.t. (~, tdeg(am_~)) is the leftmost point on the line ~. Then, I tells that the point (n, tdeg(bn . . . . )) is also on ~ and the leading terms of a~_~ and b,~_,~_~ cancel each other.

III. With the assumptions in II, assume that there exist r pairs of integers (kl, k Ÿ (k~, k'~), satisfying the following:

1. kl + kŸ . . . . . k~ + k'r, ki r kj for any i ~ j , 2. points (k i , tdeg(a~-k , ) ) and (kŸ tdeg(bn_m-k~ )) are on ~,i -- 1 , . . . , r , 3. leading temas of a m - k a b n - m - k Ÿ . . . . k" cancel one another.

This assumption is valid for the case of kl + kŸ -- t% because we have two pairs of integers (kl = 0, kŸ = ~) and (k2 ---- n, k• = 0) satisfying the above conditions. Note that , in the above assumption, the cancellation of leading terms happens in the coefficients of x ~-kl-k'~ term of AB. Note fllrther that points (ki + kŸ i ---- 1 , . . . , r , are also plotted on ~. Figure 4.2 il-

lustrates the lines ~ and ~ and the points plotted on ~.

tdeg(coeff)

A ~ ~ Fig. 4.2.


Without loss of generality, we can assume that kl < . ' . < k~, hence kŸ > �9 .. > k~r. Then, we can consider the product T -- am-k~b~-m-k~. T appears in

the coefficients of x n-k~-kŸ terna of AB, a smaller degree term than x '~-k~-kŸ and the point (kr + kŸ tdeg(T)) is also p]otted on ~, hence the leading term of T must be cance]ed by some coefficient term of AB because tdeg(T) > tdeg(fn-k~-kŸ by assumption.

By mathematical induction, this means that the cancellation of leading terms does not stop unti] we exhaust all the terms in AB, which is obviously a contra- diction, because F = A B means tcleg~ ...... (F) : tdeg~ ...... (A). tdeg~ ...... (B). This completes the proof. �9

Example 4.1.

x 6 + (4ya _ 4y2)x2 _ y2 = (x 3 + 2yx 2 + 2y2x + y)(x3 _ 2yx 2 + 2y2x _ y).

In this example, many terms cancel in the product of factors. The factors axe coeff- bounded by (0,1,2,1) w.r.t, x, and we have

(El , E 2 , . . . , E6) = (0, 0, 0, 4, 4, 4),

(El, E2, . . . ,E6) = (1 ,2 ,3 ,4 ,5 ,6) .

We see that the theory predicts the optimal coefficient bound. �9

Example 4.2.

x 6 + 2 y x 5 + y 2 x 4 _ y6x2 _ 2yTx _ y8

= (X3 + y x 2 _{_ y3 x + y4)(X3 + yX2 _ y3 x _ y4).

In this example, some of E l , . . . , E6 are rationals:

( E l , E 2 . . . . ,E6) = (1,2,3,6, 7,8),

(El, F-~2,..., E6) = (3/2, 3, 9/2, 6, 15/2, 9)

(1 ,3 ,4 ,6 ,7 ,9 ) .

We see that the factors are coeff-bounded by (0,1,3,4) w.r.t, x, which is exactly the same as predicted by theory. �9

5. Handling "Approximate" Roots

In this section we prove that the system (2.4), with ~:~1,... ,(~n replaced by "suitably approximated roots" ~Ÿ . . , ' �9 ~,~, respectively, also determines all the combinations of roots corresponding to irreducible polynomial factors of F uniquely. Here, the approximate root ~Ÿ is defined by expanding Vi into infinite power series in C { u , . . . , v} and discarding terms in ( u , . . . , v) 5+1 f o r a suitably chosen integer L. When we show the discarded temas explicitly, we write

~' = ~ / s L+I, s = ( ~ , . . . , v ) . (5.1)


~kr thermore , by solutions wi th p a r a m e t e r s u , . . . , v, we mean the solutions in which u , . . . , v are inde te rmina tes wi th no re la t ion on them. We note t ha t if sys tem (2.4) holds for nonzero Ai, 1 <: i <: n, then sys tem (3.6) has solutions wi th paxameters l i , . . . , V .

In order to invest igate the solutions of (3.6) in C { u , . . . , v} (L), we use several results of the Gr5bner basis theory of po lynomia l ideal, so we explain t h e m briefly; for details, see [1].

We consider po lynomia ls in K = C[u , . . . , v ][z l , . . . , z , , ] , and t r ea t t h e m as sums of monomia l s in this section. Let ~- be an order defined for monomials ; in our case, >- is defined for vaxiables as

Z n '�91 Z n _ 1 '2- " ' " ) - Z 1 )'- l t , . . . , V , (5.2)

and it is the lexicographic order for monomia l s in z ~ , . . . , z 1 and the to ta l -degree lexicographic order for monomia l s in u, . . . , v. T h a t is,

t / let T z ¡ z Ÿ T I Z~n ~1 = . . . = . . . z 1 , t h e n

T > T ' . = , [ ~ > ~ : ] or

[ 3 k < : n s . t . e i - - e i , i = n , . . . , k + l , a n d e k >eŸ

l e t t = u . . . . . v ~~, t ' = u ~ : . . - v ~ : , t h e n

I l t ~ - t I r > e ~ + . . . + e v ] or

[ e ~ + . . . + e . = e ~ , l + . - . + % 1 and

t is lexicographical ly higher order t h a n t'],

let W = ctT and W ' = d t 'T ' , c, �91 E C, then

W > - W ' .'. :- [ T ~ - T ' ] or [ T = T ' a n d t > - t ' ] .

Given a po lynomia l Q in K , the highest order monomia l of Q in the sense is called the head te~rn of Q and abb rev ia t ed to h t (Q) .

The GrSbner basis of a po lynomia l ideal is a special ideal basis having very useful propert ies . Given a basis of any finitely genera ted po lynomia l ideal over a field o r a Eucl idean ring, we can cons t ruc t a GrSbner basis w.r.t , any monomia l order by a s imple procedure . The cons t ruc t ion procedure wi th order >~ defined above is noth ing but the var iable el im]nation procedure , successively f rom z,, to zt , by e l iminat ing head t e r m s repeatedly. For example , the basis {G1, G2 , . . . , Gn} given in Append ix B is a Gr5bne r basis of ideal (G (1) , G ( 2 ) , . . . , G(n)), wi th G (i), i -- 1 , . . . , n, defined in (3.2), w.r. t , the order ~-.

Let I = ( P 1 , . . . , Po) be an ideal in K . Let F = { Q 1 , . . . , Qt} be a Gr5bner basis of I w.r.t , the order >-, then we have the followings:

(a) ( P l , . . . , P s ) = ( Q I , . . . , Q t ) , (5.3) (b) every solution to zl of the sys tem {P1 = 0 , . . . , P~ -- 0} in a

solut ion of {QŸ = 0 , . . . , ~r = 0}, where

362

The solutions of (3.2)

T. SASAKI, T. SAITO and T. H1LANO

{QŸ = { Q ~ , . . . , Q t } n C[u, . . . ,v][z , ] , (c) system {PI = 0 , . . . , P8 : O} has no solution with parame-

ters u , . . . , v F contains element Q(0) such that Q(0) c C [ u , . . . , vi,

(d) system {PI = O, . . . ,P8 = O} has ¡ many solutions with parameters u , . . . , v

F contains no element in C[u , . . . , v ] and for each i E {1 , . . . , n}, _F contains ah element Q(i) such that ht(Q(')) = tz~ ~, t E C[u , . . . , v ] , k, > O.

We first consider the solutions of (3.2) in C { u , . . . , U} (L).

LEMMA 5.1. Let L be a positive integer and let qa~ = ~j / S L+ I, j = 1 , . . . , n. in C { u , . . . , v } (L) a ~

{(z] = ~ ~ l , . . . , z n = ~ Ÿ (5.4)

Proo f . System (3.2) is equivalent to (3.4), and (3.4) gives

(z - Z l ) " " ( z - zn ) - z n + f n _ l z n - 1 + ' ' ' -~- .fo ~- F ( z ) (mod sL+I) .

This shows that (5.4) &re the solutions of (3.2) in C { u , . . . , v} (L). �9

We note that the defining polynomial for ~ Ÿ ~" is F ( Z l ) / S L+I and F(Zl) is contained in the Gr5bner basis w.r.t, the order ~-, of the ideal generated by the polynomials in (3.2).

Next, let us con~~ider the Gr6bner basis of the ideal

I = ( G 0 ) , . . . , G (~), H(1) , . . . , Hk)), (5.5)

where G (~) and H (~) &re defined by (3.2) and (3.5), respectively. Let the GrSbner basis be F, then either F = {1} or F is of the form

1~ = { G n , l ( Z n , . . . , z 1 ) , . . . , G l , l ( Z l ) , . . . , G l , m l ( Z l ) , G o , 1 , . . . , G o , m o } , (5.6)

where

G i , j ( z i , . . . , z 1 ) c C [ ' � 9 1 1 9 1 j = 1 , . . . ,77�91 i = 1, . . . ,n ,

G0,j E C[u , . . . , v ] , j = 1 , . . . ,m0 . (5.7)

The case F -- {1} corresponds to the case that the combined system (3.6) has no solution for any values of u , . . . , v, hence we disc&rd this case. When F is given by (5.6), the above property (a) shows that system (3.6) is equivalent to the system

{ G n , l ( Z n , . . . , z l ) --- O , . . . , G 1 , l ( Z 1 ) -- O , . . . , G l , m l ( Z l ) = O,

Go,1 = O, . . . ,Gl ,~o = O.


The Gl,j(z l) , j = 1 , . . . ,m i , are defining polynomials for the solutions to zl modulo G0,1, . . . , G0,,~0, and the above property (b) shows that there are no other defining polynomials. Hence, the case of F given in (5.6), with nonzero GI,j, corresponds to the case that (3.6) has solutions with constraints G0,j -- 0, j -- 1 , . . . , m o , on u , . . . , v. Fhrthermore, if mo = 0, which means that there is no G0,j in F, then (3.6) has solutions with parameters u , . . . , v.

LEMMA 5.2. I f the combined system (3.6) has solutions with parameters u , . . . , v in C { u , . . . , v } (L), then we have

a0, j ~ 0 (mod sL+I), j • 1 , . . . , mo, (5.8)

~h~,~ C0,~ , d~~,~d i~ (5.6) ~th (5.7). F~~th~,~o,~, ii L > ~~ th~~

Gl, j (z l ) = [some polynomial factor of F(Zl)], j = 1 , . . . , mi . (5.9)

Proof. The first c]~im of the lemma is a direct consequence of the property (d) mentioned above.

As noted above, G1,1, �9 �9 �9 Gl,ml are defining polynomials for the solutions to Zl modulo G0,1, �9 �9 �9 G0,,~o. On the other hand, the solutions of (3.6) must satisfy (3.2), and F(z l ) is the defin�91 polynomial for the solutions to Zl of (3.2). Hence, GI,j (Zl) is equal to a polynomial divisor of F(Zl) modulo G0,1, . . . , G0,,~o. Therefore, ir we have (5.8) with L > E,~, then Gl, j (z l ) must be a polynomial divisor of F(Zl), because the polynomial divisor of F is coeff-bounded by (0, E l , - . . , En) and not affected by Go,1, . . . , Go,mo. �9

THEOREM 5.3. Let (n + 1)-tuple (0, [~1,. . . , E,~) be defined by Del. 4.3. Let L be ah integer such that L >_ E,~ and h i , . . . , h,~ be polynomials in u , . . . , v satisfying tdeg(h~) _< E:~, i = 1 , . . . , n. Then, Theorem 3.2 is valid ir we replace (2.2) by

{ "~l~Ÿ "-~ "~n(~Pn 7_ hl (mod sL+I),

,k ,k (mod sL+I) ,

.~V). where ~~ = ~t~i/8 T M with S = (u, . .

(5.10)

Proof. Vandermonde's matrix formed by the coefficients of the left hand side of equations in (5.10) is non-singular modulo S T M for any natural number L. On the other hand, Lemm~ 5.2 shows that the defining polynomial for the solutions to Zl must be one of the polynomial factors of F(Zl). Hence if the system (5.10) holds then we have other systems which are the same forro as (5.10) except that the roots of each irreducible factor of F(z l ) axe systematically permuted in a suitable way. Therefore, the theorem is proved by the same reasoning as that for Theorem 3.2.

As shown in Appendix A, gi defined by (3.1) satisfies

tdeg(g,) < Ei, i = 1 , . . . , n . (5.11)


Since hi defined by (2.7) is constructed from the coefficients of F1 , . . . ,F~ which are polynomial factors of F, Theorem 4.2 shows that

tdeg(hj) < Ej , j = 1 , . . . , n . (5.12)

Hence, no solution of the combined system (3.6) contradicts with the degree con- straint on hi in Theorem 5.3. In other words, the solutions of (5.10) give all the combinations of roots corresponding to irreducible polynomial factors of F.

6. C o m p l e t e n e s s of the Algor i thm

Let f be a polynomiaJ in u , . . . ,v. By LfJ~, we denote a polynomial which is obtained by summing all the terms of f , of total-degrees not less than e. Further- more, by [f]~~, we denote the sum of al] the terms of f , of total-degrees not less than el and not greater than e2. Using (5.12), we can rewrite (5.10) as

/ ! L A I ~ i + . . . + A . ~ . J ~ l + l -- 0

Ik Ik LAl~l + ' " + A ~ ~ ~ J k ~ + l - - 0

(mod sL+I),

(mod sL+l). (6.1)

LEMMA 6.1. So long as L > Ek, systems (5.10) and (6.1) ate equivalent to each other.

Proof. System (6.1) is obviously derived from (5.10). Conversely, suppose we have relations (6.1) for a set of numbers A l , . . . , Ah. This means

Alfil i + . . - + ) ~ , , ~ ~ -- hŸ (mod sL+I),

tdeg(hŸ i - - 1 , . . . , k .

Thus we obtain (5.10). �9

The algorithm proposed by Sasaki and others [2] is based on the relations in (6.1). Using (6.1) i n s t~d of (5.10), we need not handle h,, i -- 1 , . . . , n , which are unknown polynomials in (5.10). The algorithm forros a numerical n-row matrix Mt for each set {~Ÿ . . . ,~~}, t e {1 , . . . ,k}, the ~ , j ) - t h element of which is the numerical coefficient of a monomial Ti, tdeg (Ti) > E t + 1, of ~~t. Then, it constructs a matrix

M ( t l , . . . , t k ) = tlMz + ' " + tkMk, (6.2)

where ti, �9 �9 �9 t~ are parameters. Let the row vectors of M be vi, i -- 1, �9 �9 �9 n, then solving (6.1) is equivalent to finding a set of numbers ~1 , . . . , A,~ satisfying a linear relation Alvl + " " + A,v,~ = 0. The algorithm calculates the linear relations by Gaussian e]imination.


THEOREM 6.2. When F is exactly factorized as in (2.6), the factorization al9orithm prvposed by Sasaki and others [2] /s complete, provided that the numerical coefficients ate evaluated exactly.

Proof. Let k = n in (6.1) and L = En. Then, sinee (5.10) and (6.1) are equivalent to each other as Lemma 6.1 shows, Theorem 5.3 assures tha t the number of linearly independent linear relations among row vectors of M(Q, . . . , t a ) is r, the ntunber of irreducible factors of F . Then, Theorem 5.2 in [2] assures tha t we can calculate the combinations of roots corresponding to MI the irreducible factors

of F. �9

REMARK. ID. many actual cases, we can perform the factorization by the system (6.1) with k = 1 or 2 and L = ~]1 -t- d or E2 + d, with d so chosen tha t the matr ix M1 or taMa + t2M2 has at least n nonnull columna; the cases where we need M3 or more are quite rare. Therefore, the approximate facto¡ can be performed efficiently in most practical cases. This point ~ be clarified to some extent in a coming paper.

Finally, let us mention about the approximate factorization. Let F(x, u , . . . , v) be monic w.r.t, x and F(x, 0 , . . . , 0) be approximately square-free with accuracy greater than O(r where e i s a small positive number. We assume tha t F is not exactly factorized but approximately factorized as

F = Fa.. . Fr + 6F, mmc(SF)/mmc(F) = O(e), (6.3)

where Fa, . . . , F~, 5F C C [ u , . . . , v][x]. Let F1, . . . , F~ be expressed as in (2.6). Then, Theorem 3.2 is valid if we replace F by F - 6F, hence Theorem 5.3 is also valid by the same replacement.

Now, let F be factorized over C { u , . . . , v} as

F(x) = (x - ~ a ) " " " (x -- ~ n ) , (6.4)

~ j - - * ~ j when 6F---*0, j = l , . . . , n .

Let 5ha,... ,Shk be defmed as

[ )~ -_, 1Ek+d _ 6hl [~l~Ÿ + "~-J~1+1

^ ) t , ~ t k l E k + d ~ 5hi [~a~Ÿ + ,,W,, Jt,+a

(mod sL+a),

(mod S L + a ) ,

(6.5)

where ~~ = ~i /S L+a �91 S etc. are the same as those in Eq. (6.1). Then, since ~j is approximately equal to ~j , j = 1 , . . . , n, we have

mmc(~h,) ~ max{l~al,...,[~~l} x O(~), i = l , . . . , k . (6.6)

Therefore, by finding the approximately dependent linear combinations

�9 ~1V1 ~- ' ' " - [ - ,,~nVn = max{[Al[ , . . . , I)~,~l} x O(~) x 1,


with v i , . . . , vn defined below (6.2), we can find the app rox ima te factors. Thus, we obta in the follow~ing.

THEOREM 6.3. The approximate factorization algorithm proposed by Sasaki and others [2] /s complete, provided that the numerical coej~cients ate calculated with ah enough accuracy.

Acknowledgement. The au thors t h a n k Dr. K. Yokoyama of Fuj i tsu Labs. I IAS for valuable discussions and commen t s on the relat ions among the roots.

A. On Systems (3.2) and (3.4)

We consider the following two sys tems (f~ = ( - 1 ) ' f n - , , i -- 0 , . . . , n - 1).

I Zl -t- z2 -t- . . . + zn =- gl, 2 Z~+Z~+'''+Zn=~,

n n n Z l -~- Z2 -~- " " " ~- Zn = gn"

(A.1)

I Z I ~ _ Z 2 _ . ~ . . . _ . I . . Z n = f Ÿ Z l Z 2 "~ . . . ~- Z l Z n ~- . . . --~ Z n _ l Z n ~-- f ~ ,

Z l Z 2 . * , z n ~- f z n .

(A.2)

As is well-known, g~ and fŸ i = 1 , . . . , k (k < n), satisfy the following ident i ty (Newton 's formula):

gk -- g k - l f Ÿ -{- gk-2f2 . . . . ~- (--1)k-lgl f~_ 1 ~- ( - -1)kkf~ = 0. (A.3)

By this, we can express f Ÿ f~ successively in t e rms of g l , . . . , gk . Conversely, g l , - -- , gk can be expressed in t e rms of f Ÿ f~, because g l , - . . , gk �91 symmet r i c polynomials . Therefore , sys tem (A.1) is equivalent to sys tem (A.2).

hn - the rmore , (A.3) shows tha t if we expand g~ in t e rms of f Ÿ l i then the expansion coefficients are integers.

Now, assume t h a t g~ and f~, i = 1 , . . . , n , �91 polynomials in u , . . . ,v, and pu t

ei = tdeg(fŸ i = l , . . . , n .

Using (A.3), we ob ta in upper bounds of tdeg(g~), i = 1 , . . . , n, as follows.

tdeg(gl ) = el -- E l ,

tdeg(g2) _< max{e2, el + E l } = max{e2, 2el} - E2,

tdeg(gn) < max{en , en-1 + E l , e n - 2 + F__~,... ,e l + En_l}

= max{en , en-1 + et , e~-2 + e2, en-2 + 2 e l , . . �9 he1} --= En,


where the e lements of t he set for Ei r ange over all the pa r t i t i ons of in teger i.

B . R e d u c t i o n o f S y s t e m (3 .2)

In th is append ix , we conver t the sy s t em (3.2), or equivalent sy s t em (3.4), into

the form

{Gl (Z l ) = 0, G 2 ( z 2 , z l ) = 0 , . . . , G n ( z n , Z n _ l , . . . , Z l ) = 0}. (B.1)

For convenience, we express the coefficients of F as

F ( x ) z n .! n - 1 --! n--2 = - .71 x + ] 2 x . . . . + ( - 1 ) ~ f : . (8 .2)

We denote the n roo t s of F a s ~ a , . . . ,~,~ as before.

Accord ing to L e m m a 3.1, sys tem (3.2) has n! solut ions given in (3.3). This

means t h a t zx has n different so lu t ions ~ 1 , . . . , ~n. Hence, G l ( Z l ) mus t be

G l ( z x ) = (Zl - r161 (Zl - ~q = F ( Z l ) .

Next , for Zl = qoi, 1 < i < n, z2 has n - 1 different so lu t ions ~ 1 , . . - , ~ i - 1 ,

~i+1, �9 . . , ~~. T h a t is,

j = l , r

n--1 er n- -2 et~ n- -3 --z2 +(~~-:1~z2 +(~~-~JI+J2~z~ + . . . .

This equa l i ty is val id for i = 1 , . . . , n, hence we have 1

n - 1 et, ,~-2 ' . ' , , - 3 (B.3) G 2 ( z 2 , z l ) = z 2 ~-(Zl - - J l ) Z 2 " + ' ( z 2 - - Z l f l + ] 2 ) z 2 @ " ' .

Similaxly, we have

n--2 et', n--3 G3(z3, z2, Zl) : z 3 --I- (z2 -t- Zl - l i ) z 3

e t ~ n - - 4 + (z~ + z : z l + Z~l - z : f Ÿ - z l f Ÿ + h : z 3 + . . . ,

G n _ l ( Z n _ l , " , Z l ) 2 t �9 . = Z n _ 1 -[- ( Z n - 2 - [ - ' ' " -[- Z l -- f l ) Z n - 1 (B.4)

2 + z 2 z t + (Zn-- 2 + Z n - - 2 Z n - - 3 + . . . . n - 2 f l - -

. . . . z~fŸ + f;),

G n ( z n , . . . , Z l ) = z n ~- ( Z n _ 1 ~ - ' " ~- z I - - f Ÿ

I t is in te res t ing to po in t out t h a t Ga , . �9 �9 G,, a te ob t a ined by s u b s t i t u t i n g the

lower degree po lynomia l s into a h igher one, successively, in the sy s t em (3.2), or

1 The following expressions were suggogted by Dr. Z. Liu of Institute of System Sciences, Beijing, China, although the authors derived them by themselves.


(A.1). For example , s u b s t i t u t i n g z,~ = gl - (z1 + "'" ~- Zn--1) i n to G (2) a n d us ing

gl = f Ÿ aJld g2 = f~2 _ 2f~, we obt�91 Gn_l(Zn_l~... ,Z l ) . Needless to say, s y s t e m (3.2) is e q u i v a l e n t to s y s t e m (B.1), because b o t h sys-

t e m s have t h e s a m e ,set of so lu t ions .

References

[1] B. Buchberger, Ah algorithmic method in polynomial ideal theory. Multidimeasional Sys- tems Theory (ed. Booe, N.K.), Reidel, Dordrecht, 1985.

[2] T. Sasaki, M. Suzuki, M. Kol• and M. Sasaki, Approximate factorization of multivariate polynomials and absolute irreducibility testing. Japan J. Indust. Appl. Math., 8 (1991), 35~375.

[3] B.L. van der Waerden, Moderne Algebra. Springer Verlag, Berlin, 2nd ed., 1937.

analysis of approximate factorization algorithm i

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