Table Of ContentMultivariate Polynomial Factorization by
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0 Interpolation Method ∗
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n
a
Jingzhong Zhang Yong Feng
J
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2 Xijin Tang
Laboratory for Automated Reasoning and Programming
]
G Chengdu Institute of Computer Applications
A Chinese Academy of Sciences
. 610041 Chengdu, P. R. China
h
t E-mail: [email protected], [email protected]
a
m
[
1
v Abstract
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7
Factorizationofpolynomialsarisesinnumerousareasinsymboliccom-
6
putation. It is an important capability in many symbolic and algebraic
1
computation. There are two type of factorization of polynomials. One is
0
7 convention polynomial factorization, and the other approximate polyno-
0 mial factorization.
/ Conventionalfactorizationalgorithmsusesymbolicmethodstogetex-
h
actfactorsofapolynomialwhileapproximatefactorizationalgorithmsuse
t
a numericalmethodstogetapproximatefactors ofapolynomial. Symbolic
m computationoften confrontintermediateexpressionswell problem,which
: lowertheefficiencyoffactorization. Thenumericalcomputationisfamous
v for its high efficiency, but it only gives approximate results. In this pa-
i
X per,wepresentanalgorithm whichuseapproximatemethodtogetexact
factors ofamultivariatepolynomial. Compared withothermethods,this
r
a method has the numerical computation advantage of high efficiency for
some classof polynomialswith factors oflower degree. Theexperimental
results show that the method is more efficient than factor in Maple 9.5
for polynomials with more variables and higher degree.
Key words: Factorization of multivariatepolynomials, Interpolation
methods, Numerical Computation, Decomposition of AffineVariety.
∗TheworkispartiallysupportedbyChina973ProjectNKBRPC-2004CB318003.
1
1 Introduction
Polynomialfactorizationplaysasignificantroleinmanyproblemsincludingthe
simplification, Gr¨obnerbasis and solvingpolynomialequations etc. It has been
studied for a long time and some high efficient algorithms have been proposed.
There are two type of factorization of polynomials. One is convention polyno-
mial factorization, and the other is approximate polynomial factorization.
ThemodernconventionalfactorizationmethodsfollowZassenhaus’approach
[15][16]. First,Multivariatepolynomialfactorizationisreducedtobivariatefac-
torizationdue to Bertini’s theoremandhensellifting[5][6]. Then oneofthe two
remaining variables is specialized at random. The resulting univariate polyno-
mialisfactoredandits factorsarelifted upto a highenoughprecision. Atlast,
the lifted factors are recombined to get the factors of the original polynomial.
Approximate factorization is a natural extension of conventionalpolynomial
factorization. It uses approximate methods to get approximate factorization
of polynomial. The approximate factorization is not popular now, but there
are some papers to discuss it. In 1985, Kaltonfen presented an algorithm for
performing the absolute irreducible factorization, and suggested to perform his
algorithm by floating-point numbers, then the factor obtained is an approxi-
mate one. However, the concept of approximate factorization appeared first
in a paper on control theory[10]. The algorithm is as follows: At first express
the two factors G and H of the polynomials F with unknown coefficients by
fixing their terms, then determine the numerical coefficients so as to minimize
kF −GHk. Huang et al. pursuit this approach, but the algorithm seems to
be rarely successful, unless G or H is a polynomial of several terms. In 1991,
Sasaki et al. proposed a modern algorithm[11], which use power-series roots to
find approximate factors. This algorithm is successful for polynomials of small
degrees. Subsequently, Sasaki et al. presented another algorithm[12] which uti-
lizeszero-sumrelations. Thezero-sumrelationsarequiteeffectivefordetermin-
ing approximate factors. However, computation based on zero-sum relations
is practically very time-consuming. In [13], Sasaki, T. presented an effective
method to get as many zero-sum relations as possible by matrix operations so
that approximate factorization algorithm is improved. In [3], Corless et al pro-
posed an algorithm for factoring bivariate approximate polynomial based on
the idea of decompositionof affine variety. However,it is noteasy to generalize
the algorithmto factormultivariateapproximatepolynomials. Recently,Zhang
et al[17] proposed an algorithm for obtaining exact rational number from its
approximate floating number. In this paper, basing on the algorithm in [17],
we present an algorithm which use approximate method to get exact factors
of a polynomial. It can be regard as a generalization of Corless’ algorithm in
multivariate polynomial and exact polynomial case.
The remainder of the paper is organizedas follows. Section 2 presents a for-
2
mula by which a polynomial is constructed from sampled points on its variety.
A condition is given for the formula to determine only one polynomial up to
a nonzero constant multiples, and the error estimation is discussed. Section 3
givesareviewofamodifiedcontinuedfractionmethod,bywhichanexactratio-
nalnumbercanbeobtainedfromitsapproximation. Section4firstdiscussesthe
errorcontrol,andthenproposesafactorizationalgorithmformultivariatepoly-
nomial over rational number field. Section 5 gives some experimental results.
The final section makes conclusion.
2 Interpolation method
Polynomialinterpolationisaclassicalnumericalmethod. Itisstudiedverywell
for univariate polynomials. In general, there are four types of polynomial in-
terpolation method: Lagrange Interpolation, Neville’s Interpolation, Newton’s
Interpolation and Hermite Interpolation. Lagrange interpolation formula can
getthe interpolationpolynomialatoncefor agivensetofdistinct interpolation
pointsandcorrespondingvalues(x ,f ),i=1,···,n+1. Itisveryusefulinsome
i i
situations in which many interpolation problems are to be solved for the same
setofinterpolationpointsx ,i=1,···,n+1,butdifferentsetsoffunctionvalues
i
f ,i=1,···,n+1. UnlikeLagrangeinterpolationmethodwhichsolvetheinter-
i
polation problem all at once, Neville’s interpolation method solve the problem
for smaller sets of interpolation points first and then update these solutions to
obtainthesolutiontothefullinterpolationproblem. Itaimsatdeterminingthe
value of the interpolating polynomial at some point. It is less suited for deter-
mining the interpolating polynomial. If the interpolatingpolynomialis needed,
Newton’s Interpolation formula is preferred. Just like Neville’s interpolation
method, it first get interpolating polynomial for smaller sets of interpolation
points and then update the polynomial for a larger sets of interpolationpoints,
step by step, and finally, the interpolating polynomial is obtained for the set
of the full interpolation points. If the interpolating problem prescribes at each
interpolationpointx ,i=1,···,n+1notonlythevaluebutalsothederivatives
i
of desired polynomial, then the Hermite formula is preferred.
For univariate polynomial interpolation, n+1 distinct interpolation points
and their corresponding values determine only one polynomial with degree less
than or equal to n. However, the interpolation points for multivariate poly-
nomial interpolation can not be chosen arbitrarily. They need to satisfy some
conditions. So we need a definition as follows:
Definition 1 Let Θ be a set of n-dimension points and P a polynomial space.
We call Θ Proper interpolation points of P if for any f defined on Θ, there
is a unique polynomial p∈P matching f at Θ.
3
In definition 1, a polynomial p matching f at Θ means p|Θ = f|Θ. In
general, we can determine interpolation polynomial space such as: if knowing
the total degree d of f, we choose P = {p|deg(p) ≤ d ,p ∈ K[x ,x ,···,x ]};
1 2 n
if knowing the degree di of f in xi(i = 1,···,n), we choose P = {p|∧i=1,2,···,n
(deg (p)≤d ),p∈K[x ,x ,···,x ]}. Oncetheinterpolationpolynomialspace
xi i 1 2 n
isdetermined,theproperinterpolationpointsΘofP canbesetbyinterpolation
methods[9][1].
In this paper, we need to construct a polynomial from some points of its
variety. Values of the polynomial at interpolation points are all zero. So, we
introduce an interpolation formula for this case.
Let f(x ,x ,···,x ) be a polynomialto be interpolated. It is representedas
1 2 n
follows:
f(x ,x ,···,x )=c Xα1 +c Xα2 +···+c Xαm, (1)
1 2 n 1 2 m
where Xαi = xd11,ixd22,i···xdnn,i are the distinct monomials, and ci are the cor-
responding coefficients.
Let p1,p2,···,pm−1 be points on variety of f(x1,x2,···,xn), where pi =
(p ,p ,···,p ) for i = 1,···,m − 1. Pαj = pd1,jpd2,j···pdn,j denote the
11 12 1n i i1 i2 in
value of the monomial Xαj at pi. An interpolation formula is as follows:
Xα1 Xα2 ··· Xαm
(cid:12)(cid:12) Pα1 Pα2 ··· Pαm (cid:12)(cid:12)
G(x1,x2,···,xn)=(cid:12)(cid:12) ··1· ··1· ··· ··1· (cid:12)(cid:12) (2)
(cid:12)(cid:12)(cid:12) Pmα1−1 Pmα2−1 ··· Pmαm−1 (cid:12)(cid:12)(cid:12)
(cid:12) (cid:12)
Next,weneedtoknowwhatconditiontheinterpolationpointsp1,p2,···,pm−1
shouldsatisfysoastoensureformula(2)todetermineauniquepolynomialand
it is f(x ,x ,···) up to a nonzero constant multiplies.
1 2
Let X∗ denote the minor of Xαi resulting from the deletion of row 1 and
i
column i in formula (2). We have a theorem as follows.
Theorem 1 Let f(x ,x ,···,x ) be a nonzero polynomial and it is expressed
1 2 n
asinequation(1). Ifthem−1zeroesoff(x ,x ,···,x )satisfyaconditionthat
1 2 n
G(x ,···,x )6=0informula2,thenformula(2)determinesauniquepolynomial
1 n
and it is polynomial f(x ,x ,···,x ) up to a nonzero constant multiples.
1 2 n
Proof: Due to f(x ,x ,···,x ) 6= 0, the coefficients of f(x ,x ,···,x ) are
1 2 n 1 2 n
not all zero. Assume that c 6= 0,c 6= 0,···,c 6= 0, and their corresponding
monomials are Xαi1,Xαi2,·i1··,Xαisi2. Let Xi∗1,Xisi∗2,···,Xi∗s denote the minors
of Xαi1,Xαi2,···,Xαis in formula (2) respectively.
∗ ∗ ∗
First,weshowthatifoneofX ,X ,···,X isnonzero,thenf(x ,x ,···,x )=
i1 i2 is 1 2 n
cG(x ,x ,···,x ),where c is a nonzeroconstant. Without loss ofgenerality,let
1 2 n
4
∗
us assume that c 6=0 and X 6=0. So we have
k k
G(x ,x ,···,x )
1 2 n
Xα1 Xα2 ··· Xαk−1 Xαk Xαk+1 ··· Xαm
(cid:12)(cid:12) Pα1 Pα2 ··· Pαk−1 Pαk Pαk+1 ··· Pαm (cid:12)(cid:12)
= (cid:12) 1 1 1 1 1 1 (cid:12)
(cid:12) ··· ··· ··· ··· ··· ··· ··· ··· (cid:12)
(cid:12)(cid:12)(cid:12) Pmα1−1 Pmα2−1 ··· Pmαk−−11 Pmαk−1 Pmαk−+11 ··· Pmαm−1 (cid:12)(cid:12)(cid:12)
(cid:12) Xα1 Xα2 ··· Xαk−1 ckXαk Xαk+1 ··· X(cid:12)αm
= 1 (cid:12)(cid:12)(cid:12) P1α1 P1α2 ··· P1αk−1 ckP1αk P1αk+1 ··· P1αm (cid:12)(cid:12)(cid:12)
c (cid:12) ··· ··· ··· ··· ··· ··· ··· ··· (cid:12)
k (cid:12)(cid:12)(cid:12) Pmα1−1 Pmα2−1 ··· Pmαk−−11 ckPmαk−1 Pmαk−+11 ··· Pmαm−1 (cid:12)(cid:12)(cid:12)
(cid:12) (cid:12)
For i = 1,2,···,k−1,k+1,···,m, addition of c times column i to column k
i
yields:
G(x ,x ,···,x )
1 2 n
= 1 (cid:12)(cid:12)(cid:12) PX1αα11 ······ PX1ααkk−−11 PPkiki==11cciiPX1ααkk PX1ααkk++11 ······ PX1ααmm (cid:12)(cid:12)(cid:12)
c (cid:12) ··· ··· ··· ··· ··· ··· ··· (cid:12)
k (cid:12) (cid:12)
(cid:12) Pα1 ··· Pαk−1 k c Pαk Pαk+1 ··· Pαm (cid:12)
(cid:12)(cid:12) m−1 m−1 Pi=1 i m−1 m−1 m−1 (cid:12)(cid:12)
Xα1 ··· Xαk−1 f(x1,···,xn) Xαk+1 ··· Xαm
1 (cid:12)(cid:12) Pα1 ··· Pαk−1 0 Pαk+1 ··· Pαm (cid:12)(cid:12)
= (cid:12) 1 1 1 1 (cid:12)
c (cid:12) ··· ··· ··· ··· ··· ··· ··· (cid:12)
k (cid:12)(cid:12)(cid:12) Pmα1−1 ··· Pmαk−−11 0 Pmαk−+11 ··· Pmαm−1 (cid:12)(cid:12)(cid:12)
(−1(cid:12))k+1 (cid:12)
∗
= X f(x ,···,x )
c k 1 n
k
∗
Duetoc 6=0andX 6=0,itfollowsthatf(x ,x ,···,x )=cG(x ,x ,···,x ),
k k 1 2 n 1 2 n
where c= (−1)k+1ck is nonzero constant.
X∗
k
∗ ∗ ∗
Second, we assert that one of X ,X ,···,X must be nonzero. We prove
i1 i2 id
∗ ∗ ∗
it by contradiction. Let us assume that X =0,X =0,···,X =0.
i1 i2 id
Under the assumption of the theorem, G(x1,x2,···,xn) = Pmi=1biXαi 6= 0
and f(x ,x ,···,x ) 6= 0. So, not all of their coefficients are zero. Assume
1 2 n
that b 6= 0,b 6= 0,···,b 6= 0 and c 6= 0,c 6= 0,···,c 6= 0. Since
∗ h1 ∗ h2 ∗ hz i1 i2 is
X = 0,X = 0,···,X = 0, it holds that i 6= h for k = 1,···,s and
i1 i2 is k j
j =1,···,z. Hence,wehavethatXαik 6=Xαhj fork =1,···,sandj =1,···,z.
Let
H(x ,x ,···,x ) = f(x ,x ,···,x )+G(x ,x ,···,x )
1 2 n 1 2 n 1 2 n
= ci1Xαi1 +···+cisXαis +bh1Xαh1 +···+bhzXαhz
6= 0
5
∗
Because of b 6=0 and X 6=0, it has been shown above that
h1 h1
G(x ,x ,···,x )=c¯H(x ,x ,···,x ),
1 2 n 1 2 n
where c¯is nonzero constant. Hence we deduce that
G(x ,x ,···,x )−c¯H(x ,x ,···,x )≡0
1 2 n 1 2 n
. However it is impossible because the term c¯ci1Xαi1 6= 0 is not monomial of
∗ ∗ ∗
G(x ,···,x ). ThereforeweshowthatoneofX ,X ,···,X mustbenonzero.
1 n i1 i2 is
The proof of the theorem is finished.
Theabovetheoremshowsthatifformula(2)givesanonzeropolynomial,and
the first row contains all monomials of an interpolating polynomial. then it is
the interpolating polynomial up to nonzero constant multiples.
However,due to floating-pointcomputation, we only getapproximatezeroes
off. Accordingly,we only obtainapproximatefactors. Inthe remainingofthis
section,westudyhowtheerrorresultingfromfloatingcomputationaffectsthat
of factors. For simplicity, let us give a definition:
Definition 2 Let Xαj = xαj,1xαj,2···xαj,n, where x ,...,x are complex
i i,1 i,2 i,n i,1 i,n
number and α ,···,α are nonnegative integer. A generalized Vandemonder
j,1 j,n
determinant is defined as follows:
Xα1 Xα2 ··· Xαm
(cid:12)(cid:12) X1α1 X1α2 ··· X1αm (cid:12)(cid:12)
Vm =(cid:12)(cid:12) ··2· ··2· ··· ··2· (cid:12)(cid:12) (3)
(cid:12) (cid:12)
(cid:12) Xα1 Xα2 ··· Xαm (cid:12)
(cid:12) m m m (cid:12)
(cid:12) (cid:12)
We have an estimation of generalized vandemonder determinant as follows:
Theorem 2 Let M = max {Xαj} and B = max {kXαj −Xαkk,|Xαi −
i,j i i,j,k i i j
Xαik}. Then for m≥2 it holds that
k
|V |≤m!Mm−1B
m
Proof: We prove it by inductive method. When m = 2, the generalized Van-
demonder determinant is
Xα1 Xα2
V2 = (cid:12)(cid:12) X1α1 X1α2 (cid:12)(cid:12)=X1α1X2α2 −X2α1X1α2
(cid:12) 2 2 (cid:12)
= (cid:12)Xα1Xα2 −Xα(cid:12)1Xα2 +Xα1Xα2 −Xα1Xα2
1 2 1 1 1 1 2 1
= Xα1(Xα2 −Xα2)+Xα2(Xα1 −Xα1)
1 2 1 1 1 2
So,
|V |≤|Xα1(Xα2 −Xα2)|+|Xα2(Xα1 −Xα1)|≤2MB =2!M2−1B
2 1 2 1 1 1 2
6
Assume that |V | ≤ m!Mm−1B for m = k. Let us show that it holds for
m
m=k+1. We expand V by minors as follows
k+1
k+1 k+1
|V |=| (−1)i+jXαjVi,j|≤ |Xαj|∗|Vi,j|
k+1 X i k X i k
j=1 j=1
According to our assumption that |Vi,j|≤k!Mk−1B, we have
k
k+1 k+1
|V |≤ |Xαj|∗|Vi,j|≤ M ∗k!Mk−1B =(k+1)!MkB
k+1 X i k X
j=1 j=1
The proof is finished.
Theorem 3 Let M = max {Xαj} and B = max {kXαj −Xαkk,|Xαi −
i,j i i,j,k i i j
Xαik} and ε=maxm |a |. A determinant is as follows:
k i=1 i
Xα1 Xα2 ··· a ··· Xαm
Vm =(cid:12)(cid:12)(cid:12)(cid:12) X··12α·1 X··12α·2 ······ ··a·12··· ······ X12αm (cid:12)(cid:12)(cid:12)(cid:12)
(cid:12) (cid:12)
(cid:12)(cid:12) Xmα1 Xmα2 ··· am ··· Xmαm (cid:12)(cid:12)
(cid:12) (cid:12)
Then we have an estimate that |V |≤Mm−2m!Bε for m≥3.
m
Proof. Expanding V by column (a ,···,a )T and then using theorem 2, we
m 1 m
can get the proof.
And now, we study the difference between two generalized Vandemonder
determinants.
Theorem 4 Let
Xα1 Xα2 ··· Xαm
(cid:12)(cid:12) X1α1 X1α2 ··· X1αm (cid:12)(cid:12)
V(1) =(cid:12) 2 2 2 (cid:12)
m (cid:12) ··· ··· ··· ··· (cid:12)
(cid:12) (cid:12)
(cid:12) Xα1 Xα2 ··· Xαm (cid:12)
(cid:12) m m m (cid:12)
(cid:12) (cid:12)
and
Yα1 Yα2 ··· Yαm
(cid:12)(cid:12) Y1α1 Y1α2 ··· Y1αm (cid:12)(cid:12)
V(2) =(cid:12) 2 2 2 (cid:12)
m (cid:12) ··· ··· ··· ··· (cid:12)
(cid:12) (cid:12)
(cid:12) Yα1 Yα2 ··· Yαm (cid:12)
(cid:12) m m m (cid:12)
and assume that M =max {(cid:12)Xαj,Yαj} , B =max (cid:12){kXαj −Xαkk,|Xαi −
i,j i i i,j,k i i j
Xαik,kYαj −Yαkk,|Yαi −Yαik} and ε = maxm {kXαj −Yαjk}. Then it
k i i j k i,j=1 i i
holds for m≥3 that
|V(1)−V(2)|≤mm!Mm−2Bε
m m
7
Proof:
Xα1 Xα2 ··· Xαm Yα1 Yα2 ··· Yαm
(cid:12)(cid:12) X1α1 X1α2 ··· X1αm (cid:12)(cid:12) (cid:12)(cid:12) Y1α1 Y1α2 ··· Y1αm (cid:12)(cid:12)
V(1)−V(2) =(cid:12) 2 2 2 (cid:12)−(cid:12) 2 2 2 (cid:12)
m m (cid:12) ··· ··· ··· ··· (cid:12) (cid:12) ··· ··· ··· ··· (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) Xα1 Xα2 ··· Xαm (cid:12) (cid:12) Yα1 Yα2 ··· Yαm (cid:12)
(cid:12) m m m (cid:12) (cid:12) m m m (cid:12)
Xα1 Xα2(cid:12) ··· Xαm Yα1 X(cid:12) α2(cid:12) ··· Xαm (cid:12)
(cid:12)(cid:12) X1α1 X1α2 ··· X1αm (cid:12)(cid:12) (cid:12)(cid:12) Y1α1 X1α2 ··· X1αm (cid:12)(cid:12)
=(cid:12) 2 2 2 (cid:12)−(cid:12) 2 2 2 (cid:12)
(cid:12) ··· ··· ··· ··· (cid:12) (cid:12) ··· ··· ··· ··· (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) Xα1 Xα2 ··· Xαm (cid:12) (cid:12) Yα1 Xα2 ··· Xαm (cid:12)
(cid:12) m m m (cid:12) (cid:12) m m m (cid:12)
(cid:12) Yα1 Xα2 ··· Xαm (cid:12) (cid:12)Yα1 Yα2 ··· Xαm (cid:12)
(cid:12)(cid:12) Y1α1 X1α2 ··· X1αm (cid:12)(cid:12) (cid:12)(cid:12) Y1α1 Y1α2 ··· X1αm (cid:12)(cid:12)
+(cid:12) 2 2 2 (cid:12)−(cid:12) 2 2 2 (cid:12)
(cid:12) ··· ··· ··· ··· (cid:12) (cid:12) ··· ··· ··· ··· (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) Yα1 Xα2 ··· Xαm (cid:12) (cid:12) Yα1 Yα2 ··· Xαm (cid:12)
(cid:12) m m m (cid:12) (cid:12) m m m (cid:12)
+(cid:12)··· (cid:12) (cid:12) (cid:12)
Yα1 Yα2 ··· Xαm Yα1 Yα2 ··· Yαm
(cid:12)(cid:12) Y1α1 Y1α2 ··· X1αm (cid:12)(cid:12) (cid:12)(cid:12) Y1α1 Y1α2 ··· Y1αm (cid:12)(cid:12)
+(cid:12) 2 2 2 (cid:12)−(cid:12) 2 2 2 (cid:12)
(cid:12) ··· ··· ··· ··· (cid:12) (cid:12) ··· ··· ··· ··· (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) Yα1 Yα2 ··· Xαm (cid:12) (cid:12) Yα1 Yα2 ··· Yαm (cid:12)
(cid:12) m m m (cid:12) (cid:12) m m m (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
Fromtheorem3,itholdsthat|V(1)−V(2)|≤m∗Mm−2m!Bε=mm!Mm−2Bε.
m m
The proof is finished.
3 Continued fraction method
Aswesaidabove,ourmethodistouseapproximatemethodtogetexactfactors
of a multivariate polynomial over rational number field. So we need to recover
the exact coefficients of a polynomial from its approximate coefficients. In this
section,weintroduceacontinuedfractionmethodtorecoverexactrationalnum-
berfromits approximation. As weknow,acontinuedfractionrepresentationof
a real number is one of the forms:
1
a + , (4)
0 a + 1
1 a2+a3+1···
wherea isaintegeranda ,a ,a ,···arepositiveintegers. Onecanabbreviate
0 1 2 3
the above continued fraction as
[a ;a ,a ,···]
0 1 2
In order to recover exact rational number, we introduce a control error into
the conventional continued fraction method. The continued fraction method is
modified as follows.
8
Algorithm 1 Continued fraction method
Input: a nonnegative floating-point number a and ε >0;
1
Output: a rational number b
Step 1: i:=1 and x :=a;
1
Step 2: Getting integral part of x and assigning it to a , assigning
i i
its remains to b . If b <ε , then goto Step 5;
i i 1
Step 3: i:=i+1;
Step 4: x := 1 and goto Step 2;
i bi−1
Step 5: Computing expression (4) and assigning it to b.
Step 6: return b.
In [17], we discussed how to get error control ε > 0. The theorem is as
1
follows:
Theorem 5 Let n /n be a reduced rational number and r its approximation.
0 1
Assumethatn ,n arepositive integersand L≥max{n ,2}. K is apositive in-
0 1 1
teger. The continuedfraction representations ofn /n andr are[a ,a ,···,a ]
0 1 0 1 N
and [b ,b ,···,b ] respectively. If |d| = |r−n /n | < 1/((2K +2)L(L−1)),
0 1 M 0 1
then one of the two statement holds
• a =b for i=0,···,N, and b ≥K;
i i N+1
• a =b for i=0,···,N −1, and b =a −1, b =1, b ≥K.
i i N N N+1 N+2
From theorem 5, getting exact non-negative number n /n from its approx-
2 1
imation r is summarized as follows:
0
Algorithm 2 Obtaining Exact Number
Step 1: estimating an upper bound of the denominator of n /n , De-
2 1
noted by L;
Step 2: computing
1
β =
(2L+2)L(L−1))
Step 3: obtaining r by approximate method such that |r −n /n |<
0 0 2 1
β;
Step 4: taking ε =1/L in algorithm 1 and calling algorithm 1 to get
1
b. So n /n =b.
2 1
9
4 Factoring Multivariate Polynomials by Approx-
imate Method
In this paper, we only discuss factorization of a multivariate polynomial over
rational number field. So its coefficients are all rational numbers. In order
to get factors of a multivariate polynomial over rational number field, we first
compute its factors over complex field. These factors are complex coefficient
polynomials. Products of some of them must be real polynomials. We get
the products which are approximate rational coefficient factors of the original
polynomial. Finally, transforming these real products into rational coefficient
polynomialsyieldsfactorsofthe originalpolynomialoverrationalnumberfield.
A set
V(f)={(a ,···,a )∈Cn :f(a ,···,a )=0}
1 n 1 n
is called affine variety of f(x ,···,x ). An affine variety V ⊂ Cn is irre-
1 n
ducible if wheneverV is writtenin the formV =V ∪V , whereV andV are
1 2 1 2
affine varieties, then either V =V or V =V.
1 2
Let f(x ,x ,···,x ) be a square free polynomial over complex number field
1 2 n
C andf =f f ···f ,wheref isdistinctirreduciblepolynomials. Then<f >
1 2 m i
is a radical ideal. It holds as follows
V(f)=V(f )∪V(f )∪···∪V(f ), (5)
1 2 m
where V(f ) are irreducible affine varieties.
i
From equation (5), if we get a point on variety of f, it must be either on
one of V(f ) or on the intersection of them. When the point is not singular
i
point, it must be on one of V(f ) and not on the intersection of two varieties.
i
Theorem 1 shows that if getting enough points in some variety of V(f ) that
i
satisfy the condition of theorem 1, we can recover the polynomial by formula
(2). Therefore, the procedure of factorization is as follows: First get a initial
nonsingular point on one variety of V(f ). And then obtain enough sampled
i
points on the same variety. Third, use formula (2) to get a factor and finally,
obtain a rational factor.
However,duetoapproximatecomputation,wefirstdiscusserrorcontrol,and
then study factorization.
4.1 Error control
Let f(x ,···,x ) be a polynomial to be factored over rational number field.
1 n
Accordingtoalgorithm3,thefirstthingweneedtodoistodetermineanupper
10
