Table Of ContentTrading Order for Degree in
Creative Telescoping
Shaoshi Chen1
Department of Mathematics
2
North Carolina State University
1
0 Raleigh, NC 27695-8205, USA
2
n Manuel Kauers2
a
J
Research Institute for Symbolic Computation
1
Johannes Kepler University
3
A4040 Linz, Austria
]
C
S
. Abstract
s
c
[ We analyze the differential equations produced by the method of creative telescoping applied
to a hyperexponential term in two variables. We show that equations of low order have high
2
degree, and that higher order equations have lower degree. More precisely, we derive degree
v
8 bounding formulas which allow to estimate the degree of the output equations from creative
0 telescoping as a function of the order. As an application, we show how the knowledge of these
5 formulas can be used to improve, at least in principle, the performance of creative telescoping
4 implementations, and we deduce bounds on the asymptotic complexity of creative telescoping
. for hyperexponentialterms.
8
0
1
Key words: Definiteintegration, Hyperexponentialterms, Zeilberger’s algorithm.
1
:
v
i
X
r
a 1. Introduction
Creative telescoping is a technique for computing differential or difference equations
satisfiedbyagivendefinitesumorintegral.Thetechniquebecamewidelyknownthrough
theworkofZeilberger(1991),whofirstobservedthatcreativetelescopingincombination
Email addresses: [email protected] (Shaoshi Chen),[email protected] (ManuelKauers).
1 Currentaddress.TheworkdescribedherewasdonewhileS.C.wasemployedaspostdocatRISCinthe
FWFprojectsY464-N18andP20162-N18.AtNCSU,S.C.issupportedbytheNSFgrantCCF-1017217.
2 M.K.wassupportedbytheFWFgrantY464-N18.
Preprint submitted to Elsevier 1 February 2012
d
60
50
40
30
20
10
0 r
0 5 10 15 20
Fig. 1. Sizes(r,d)of creativetelescoping relations for theintegral of acertain rational function
with Gosper’s algorithm (Gosper, 1978) for indefinite hypergeometric summation leads
to a complete algorithm for computing recurrence equations of definite hypergeomet-
ric sums. This algorithm is now known as Zeilberger’s algorithm (Zeilberger, 1990). In
its original version, it accepts as input a bivariate proper hypergeometric term f(n,k)
and returns as output a linear recurrence equation with polynomial coefficients satisfied
by the sum F(n) = b f(n,k). An analogous algorithm for definite integration was
k=a
given by Almkvist anPd Zeilberger (1990). This algorithm accepts as input a bivariate
hyperexponential term f(x,y) and returns as output a linear differential equation with
β
polynomial coefficients satisfied by the integral F(x) = f(x,y)dy. A summary of the
α
method of creative telescoping for this case is given in SRection 2 below. For further de-
tails,variations,andgeneralizations,consultforinstancePetkovˇseket al.(1997),Chyzak
(2000),Schneider (2005),Chyzak et al.(2009),Kauers and Paule(2011).Forimplemen-
tations, see Paule and Schorn (1995), Chyzak (1998), Koepf (1998), Schneider (2004),
Abramov et al. (2004), Koutschan (2009, 2010), etc.
The equations which can be found via creative telescoping have a certain order r and
polynomialcoefficientsofacertaindegreed.Butforafixedintegrationproblem,r andd
are not uniquely determined. Instead, there are infinitely many points (r,d) 2 such
∈ N
thatcreativetelescopingcanfindanequationoforderranddegreed.Thesepointsforma
regionwhichisspecifictotheintegrationproblemathand.Figure1showsanexamplefor
sucha region.Everypoint (r,d) inthe grayregioncorrespondsto a differentialequation
of order r and degree d which creative telescoping can find for integrating the rational
function
f(x,y)= 3x2y2+9x2y+9x2+10xy2+3xy+4x+1 3x3y3+9x3y2+x3y+3x3
(cid:16) (cid:17).(cid:16)
+7x2y3+8x2y2+5x2+8xy3+10xy2+10xy+x+5y3+10y2+5y+5 .
(cid:17)
The picture indicates that low order equations have high degree, and that the degree
decreases with increasing order. But what exactly is the shape of the gray region? And
where does it come from? And how can it be exploited? These are the questions we
address in this article.
2
How can it be exploited? There are two main reasonswhy the shape of the gray
region is of interest. First, because it can be used to estimate the size of the output
equations, and hence to derive bounds on the computational cost of computing them.
Secondly, because it can be used to design more efficient algorithms by recognizing that
some of the equations are cheaper than others.
An analysis of this kind was first undertaken by Bostan et al. (2007). They studied
the problem of computing differential equations satisfied by a given algebraic function
and found a similar phenomenon: low order equations have high degree and vice versa.
Amongotherthings,theyfoundthatanalgebraicfunctionwithaminimalpolynomialof
degreensatisfiesadifferentialequationoforderatmostnwithpolynomialcoefficientsof
degree O(n3), but also a differential equation of order 6n whose coefficients have degree
only O(n2). Their message is that trading order for degree can pay off.
The same phenomenon applies to creative telescoping, as was shown by Bostan et al.
(2010) for the case of integrating rational functions. The results in the present article
extend this work in two directions: First in that we consider the larger input class of
hyperexponential terms, and second in that we give not only isolated degree estimates
for some specific choices of r, but a curve which passes along the boundary of the gray
region and thus establishes a degree estimate as a function of the order r.
Where does it come from? The standard argument for proving the existence
of creative telescoping relations rests on the fact that linear systems of equations with
morevariablesthanequationsmusthaveanontrivialsolution.Everycreativetelescoping
relationcanbe viewedas a solutionof a certainlinear system ofequations which can be
constructedfromthe datagiveninthe input.Thereis somefreedominhowto construct
these systems, and it turns out that this freedom can be used for making the number of
variablesexceedthenumberofequations,andthustoenforcetheexistenceofanontrivial
solution.
This reasoning not only implies the existence of equations and the termination of the
algorithm which searches for them, but it also implies bounds on the output size and
on the computational cost of the algorithm. But in order to obtain good bounds, the
freedom in setting up the linear systems must be used carefully. For a good bound, we
notonlywantthatthenumberofvariablesexceedsthenumberofequations,butwealso
wantthis to happenalreadyfora reasonablysmallsystem.The shape ofthe grayregion
originates from the smallest systems which have solutions.
Verbaeten (1974, 1976) introduced a technique which helps in keeping the size of the
systems small. The idea is to saturate the linear systems by introducing additional vari-
ables in a way that avoids increasing the number of equations. We will make use of this
idea in Section 3 where we propose a design for a parameterized family of linear sys-
temswhosesolutionsgiverisetocreativetelescopingrelations.Unfortunately,itrequires
some quite lengthy and technical calculations to translate this particular design into an
inequality condition which rephrases the condition “number of variables > number of
equations” in precise terms. However, as a reward we obtain a good approximation to
the gray region as the solution of this inequality.
What is the exact shape? We don’t know. All we can offer are some rational
functions which describe the boundary of the region of all (r,d) where the ansatz de-
scribed in Section 3 has a solution (Theorem 14).The graphsof these rationalfunctions
are curves which pass approximately along the boundary of the gray region.
3
By construction, for all integer points (r,d) above these graphs we can guarantee
the existence of a creative telescoping relation of order r with polynomial coefficients
of degree d. But we have no proof that our curves are best possible. Experiments have
shown that at least in some cases, our curve describes the boundary of the gray region
exactly,or withina negligibleerror.Inothercases,there remainsa significantportionof
the gray region below our curve when r is large.
In cases where the curve from Theorem 14 is tight, we can compute the points (r,d)
for which certain interesting measures (such as computing time, output size, ...) are
minimized, as shown in Section 5. Even when the curve is not tight, these calculations
still give rise to new asymptotic bounds (including the multiplicative constants) of the
corresponding complexities. We expect that this data will be valuable for constructing
the next generation of symbolic integration software.
2. Creative Telescoping for Hyperexponential Terms
We consider in this article only hyperexponential terms as integrands. Throughout
the article, is a field of characteristic 0, and (x,y) is the field of bivariate rational
K K
functions in x and y over . Let D and D denote the derivations on (x,y) such
x y
K K
that D c = D c = 0 for all c , and D x = 1, D y = 0, D x = 0, D y = 1.
x y x x y y
∈ K
OnecanseethatD andD commute witheachotheron (x,y).We saythatafield
x y
K E
containing (x,y)isadifferential field extension of (x,y)ifthederivationsD andD
x y
K K
are extended to derivations on and those extended derivations, still denoted by D
x
E
and D , commute with each other on .
y
E
Definition 1. Anelementh ofadifferentialfieldextension of (x,y) is calledhyper-
E K
exponential (over (x,y)) if
K
D h D h
x y
(x,y) and (x,y).
h ∈K h ∈K
Whenh isahyperexponentialtermandr ,r (x,y)aresuchthat(D h)/h=
1 2 x
∈E ∈K
r and (D h)/h = r , then D D h = D D h implies D r = D r . Conversely,
1 y 2 x y y x y 1 x 2
Christopher (1999) has shown for algebraically closed ground fields that for any
K
two rational functions r ,r (x,y) with D r = D r there exist a/b (x,y),
1 2 y 1 x 2
∈ K ∈ K
c ,...,c [x,y] and e ,...,e with
0 L 1 L
∈K ∈K
L L
D c a D c D c a D c
x 0 x ℓ y 0 y ℓ
r = +D + e and r = +D + e .
1 x ℓ 2 y ℓ
c b c c b c
0 (cid:16) (cid:17) Xℓ=1 ℓ 0 (cid:16) (cid:17) Xℓ=1 ℓ
Together with Theorem 2 of Bronstein et al. (2005), it follows that there exists a dif-
ferential field extension of (x,y) and an element h with (D h)/h = r and
x 1
E K ∈ E
(D h)/h=r which we can write in the form
y 2
L
a
h=c exp ceℓ,
0 b ℓ
(cid:16) (cid:17)ℓY=1
where a [x,y], b,c ,...,c [x,y] 0 , e ,...,e , and the expressions
0 L 1 ℓ
exp(a/b) ∈andKceℓ refer to element∈s oKf on\w{hi}ch D and D∈ Kact as suggested by the
ℓ E x y
notation. We assume from now on that hyperexponential terms are always given in this
4
form, and we use the letters a,b,c ,...,c ,e ,...,e consistently throughout with the
0 L 1 L
meaning they have here.
Example 2. h=exp(x2y)√x 2y is a hyperexponential term. We have
−
D h 1+4x2y 8xy2 1
x
= − =2xy+ (x,y),
h 2x 4y 2x 4y ∈K
− −
D h x3 2x2y 1 1
y = − − =x2 (x,y).
h x 2y − x 2y ∈K
− −
For this term, we can take c =1, a=x2y, b=1, c =x 2y, e = 1.
0 1 − 1 2
Wemayadopttheadditionalcondition(withoutlossofgenerality)thatthec (ℓ>0)
ℓ
aresquarefreeandpairwisecoprime,andthate forallℓ>0.Theestimatesderived
ℓ
6∈N
belowdonotdependontheseadditionalconditions,butwilltypicallynotbesharpwhen
they are not fulfilled. For simplicity, we will exclude throughout some trivial special
cases by assuming that all e are nonzero and that max deg a,deg b + L deg c
ℓ { x x } ℓ=1 x ℓ
and max deg a,deg b + L deg c are nonzero.These latter two condiPtions encode
{ y y } ℓ=1 y ℓ
the requirement that h is nPeither independent of x nor independent of y, nor simply a
polynomial.
Applied to the hyperexponential term h, the method of creative telescoping consists
of finding, by whatever means, polynomials p ,...,p [x], not all zero, and a hyper-
0 r
∈K
exponential term Q such that
p h+p D h+ +p Drh=D Q.
0 1 x ··· r x y
An equation of this form is called a creative telescoping relation for h, the differential
operator P :=p +p D + +p Dr appearing on the left is called the telescoper and
0 1 x ··· r x
Q is called the certificate of the relation. The telescoper is required to be nonzero and
free of y, but the certificate may be zero or it may involve both x and y. When p = 0,
r
6
the number r is called the order of P, and d:=maxr deg p is called its degree.
i=0 x i
Tomotivatetheformofacreativetelescopingrelation,assumethath=h(x,y)canbe
β
interpretedasanactualfunctioninxandyandconsidertheintegralf(x)= h(x,y)dy.
α
Then integrating both sides of a creative telescoping relation implies that fRsatisfies the
inhomogeneous differential equation
p (x)f(x)+p (x)D f(x)+ +p (x)Drf(x)= Q(x,y) β .
0 1 x ··· r x y=α
(cid:2) (cid:3)
In the frequent situation that the inhomogeneous part happens to evaluate to zero, this
means that the telescoper of h annihilates the integral f.
Example 3. A creative telescoping relation for h=exp(x2y)√x 2y is
−
(3x3 6)h 2xD h=D (3x 4y)h .
x y
− − −
(cid:0) (cid:1)
ItconsistsofthetelescoperP =(3x3 6) 2xD andthecertificateQ=(3x 4y)h.For
x
− − −
thedefiniteintegralf(x):= x/2exp(x2y)√x 2ydy,weobtainthedifferentialequation
−∞
−
R
(3x3 6)f(x) 2xD f(x)=0.
x
− −
5
Creative telescoping relations for hyperexponential terms can be found with the al-
gorithm of Almkvist and Zeilberger (1990), which relies on a continuous analogue of
Gosper’s summation algorithm. A more direct approach was considered by Apagodu
(alias Mohammed) and Zeilberger (2005; 2006). After making a suitable choice for the
denominator of Q, they fix an order r and a degree s for the numerator of Q, make an
ansatz with undetermined coefficients, and obtain a linear system by comparing coef-
ficients. Appropriate choices of r and s ensure that this linear system has a nontrivial
solution, and also lead to a sharp bound on the order r of the telescoper.
Let us illustrate this reasoning for the case where the integrand is a rational function
h=u/v (x,y) with deg u<deg v and v irreducible. Fix some r. Then we have to
∈K y y
find p ,...,p (x) and a rational function Q (x,y) with
0 r
∈K ∈K
p h+p D h+ +p Drh=D Q.
0 1 x ··· r x y
A reasonable choice for Q is Q= s q yi /vr, where s=deg u+(r 1)deg v and
i=0 i y − y
q0,...,qs are unknowns, because (cid:0)wPith this c(cid:1)hoice, both sides of the equation are equal
to a rational function with the same denominator vr+1 and numerators of degree at
most deg u+rdeg v in y in which the unknowns p and q appearlinearly. Comparing
y y i j
coefficients with respect to y on both sides leads to a homogeneous linear system of at
most 1+deg u+rdeg v equations with (r +1)+(s+1) unknowns and coefficients
y y
in (x). This system will have a nontrivial solution if r is chosen such that
K
(r+1)+(s+1)>deg u+rdeg v+1 r deg v.
y y ⇐⇒ ≥ y
All these solutions must lead to a nonzero telescoper P because any nontrivial solution
with P = 0 would have a nonzero certificate Q with D Q = 0, and this is impossible
y
because s was chosen such that the numerator of Q has a strictly lower degree than its
denominator.
We have thus shown the existence of telescopers of any order r deg v. This is a
≥ y
goodbound,butitdoesnotprovideanyestimateontheirdegreesd.We willnextderive
inequalities involving both r and d by constructing linear systems with coefficients in
K
rather than in (x).
K
3. Shaping the Ansatz
Let h be a hyperexponential term and consider an ansatz of the form
r di s1 s2 h
P = p xjDi, Q= q xiyj
i,j x (cid:18) i,j (cid:19)v
Xi=0Xj=0 Xi=0Xj=0
foratelescoperP andacertificateQ.Theplanistofindagoodchoicefortheparameters
r,s ,s ,v,d ,...,d .Theonlyrestrictionwehaveisthatthelinearsystemobtainedfrom
1 2 0 r
equating all the coefficients in the numerator of the rational function (Ph D Q)/h to
y
−
zeroshouldhaveasolutioninwhichnotallthe p arezero.The remainingfreedomcan
i,j
be used to shape the ansatz such as to keep d:=maxr d small.
i=0 i
Asasufficientconditionfortheexistenceofasolution,wewillrequirethatthenumber
of terms xiyj in the numerator of the rational function (Ph D Q)/h (i.e., the number
y
of equations) should be less than r (d +1)+(s +1)(s− +1) (i.e., the number of
i=0 i 1 2
variables pi,j and qi,j). As shown iPn the following example, this condition is really just
sufficient, but not necessary.
6
Example 4. Let h = u/v be the rational function from the introduction. With r = 3,
d = d = d = d = d = 54, and Q = 62 8 q xiyj v3, comparing the
0 1 2 3 i=0 j=0 i,j
coefficients of the numerator of (Ph DyQ)/(cid:0)hPto zePro gives a lin(cid:1)e(cid:14)ar system with 787
−
variables and 792 equations. This system has a nonzero solution although 792>787.
Thisphenomenonisnotonlyanunlikelycoincidenceinthisparticularexample,butit
happenssystematicallywhentheparametersoftheansatzarenotwellchosen.Estimates
which are only based on balancing the number of variables and the number of equations
will then overshoot.It is therefore preferable to shape the ansatz for P and Q in such a
way that the linear system originating from it will have a nullspace whose dimension is
exactly the difference between the number of equations and the number of variables (or
0 if there are more equations than variables).
The goal of this section is to describe our choice for the ansatz of telescoper and
certificate.TheformoftheansatzforthetelescoperisgiveninSection3.1,thecertificate
is discussed in Section 3.2. In the beginning, we collect some facts about the rational
functions (Dih)/h which are used later for calculating how many equations a particular
x
ansatz induces. The following notational conventions will be used throughout.
Notation 5. lc p and deg p refer to the leading coefficient and the degree of the
• z z
polynomial p with respect to the variable z, respectively. For the zero polynomial, we
define deg 0:= and lc 0:=0.
∗ z −∞ z
p refers to the square free part of the polynomial p with respect to all its variables,
• e.g., (x+1)3(y+3)2 ∗ =(x+1)(y+3).Notethatp∗isonlyuniqueuptomultiplication
∗ ∗ ∗
byel(cid:0)ementsfrom (cid:1) 0 ,butthatforanychoiceofp ,thedegreesdeg p anddeg p
K\{ } ∗ x y
areuniquelydeterminedandwehavethatp (D p)/pisapolynomialinxandy.These
x
are the only properties we will use.
zn :=z(z 1)(z 2) (z n+1)and zn :=z(z+1)(z+2) (z+n 1)denote the
• − − ··· − ··· −
falling and rising factorials, respectively. For n 0 we define zn :=zn :=1.
≤
If z is a real number, then z+ :=max 0,z .
• { }
If z is a real number, then z := max x : x z , z := min x : x z ,
• and z := z+ 1 denotes ⌊the⌋ nearest{inte∈geZr to z≤. } ⌈ ⌉ { ∈ Z ≥ }
⌊ ⌉ ⌊ 2⌋
If Φ is a formula then [[Φ]] denotes the Iverson bracket, which evaluates to 1 if Φ is
•
true and to 0 if Φ is false, e.g., z+ =[[z 0]]z; δ =[[i=j]], etc.
i,j
≥
Lemma 6. Let h be a hyperexponential term and i 0.
≥
(1) If deg a>deg b, then
x x
Dih N
x = i
h c bb∗ L c i
0 ℓ=1 ℓ
for some polynomial N [x,y] with(cid:0) Q (cid:1)
i
∈K
L
∗
deg N =deg c +i deg a+deg b + deg c 1 ,
x i x 0 (cid:16) x x Xℓ=1 x ℓ− (cid:17)
L
∗
deg N deg c +i max deg a,deg b +deg b + deg c ,
y i ≤ y 0 (cid:16) { y y } y Xℓ=1 y ℓ(cid:17)
L
i
lc N =(lc c ) lc ab∗ c (deg a deg b)i.
x i x 0 (cid:16) x ℓY=1 ℓ(cid:17) x − x
7
(2) If deg a deg b, then
x ≤ x
Dih N
x = i
h c bb∗ L c i
0 ℓ=1 ℓ
for some polynomial N [x,y] with(cid:0) Q (cid:1)
i
∈K
L
∗
deg N =deg c +i deg b+deg b + deg c 1 [[ω i>ω]]δ,
x i x 0 (cid:16) x x Xℓ=1 x ℓ− (cid:17)− ∈N∧
L
∗
deg N deg c +i max deg a,deg b +deg b + deg c ,
y i ≤ y 0 (cid:16) { y y } y Xℓ=1 y ℓ(cid:17)
L
(lc c ) lc bb∗ c iωi if ω or i ω;
x 0 x ℓ
6∈N ≤
lcxNi =(lc N (cid:0) ) lc ℓbQ=b∗1 L(cid:1)c i−(ω+1)( δ 1)i−(ω+1) if ω and i>ω,
x ω+1 x ℓ
− − ∈N
(cid:0) ℓQ=1 (cid:1)
L
where ω :=deg c + e deg c and, if ω ,
x 0 ℓ x ℓ ∈N
ℓP=1
L
∗
δ :=deg c +(ω+1) deg b+deg b + deg c 1 deg N 1.
x 0 (cid:16) x x Xℓ=1 x ℓ− (cid:17)− x ω+1 ≥
Proof. All claims are proved by induction on i. For i = 0, there is nothing to show in
any of the cases. The calculations for the induction step i i+1 are as follows.
→
(1) Let v :=bb∗ L c and write m for the claimed value of deg N . Then
ℓ=1 ℓ i x i
Q
Di+1h N a L a L
x =D i c exp ceℓ c exp ceℓ
h x(cid:18)c0vi 0 (cid:16)b(cid:17)ℓY=1 ℓ (cid:19)(cid:30)(cid:18) 0 (cid:16)b(cid:17)ℓY=1 ℓ (cid:19)
∗ ∗ L
(D N )v iN D v N (D a)b ab (D b)/b N D c
x i i x i x x i x ℓ
= − + − + e
c vi+1 c vi bb∗ c vi ℓ c
0 0 0 Xℓ=1 ℓ
L L
(D N )v iN D v+N c (D a)b∗ ab∗Dxb +N v e Dxcℓ
x i − i x i ℓ x − b i ℓ cℓ
= (cid:0)ℓQ=1 (cid:1)(cid:0) (cid:1) ℓP=1 .
c vi+1
0
Since deg a>deg b by assumption, we have
x x
L
D c
x ℓ
deg (D N )v iN D v+N v e
x(cid:18) x i − i x i ℓ c (cid:19)
Xℓ=1 ℓ L
∗
deg N +deg v 1=m +deg b +deg b+ deg c 1
≤ x i x − i x x x ℓ−
L Xℓ=1
∗
<m +deg a+deg b + deg c 1=m .
i x x x ℓ− i+1
Xℓ=1
Furthermore, because of
(Dxa)b∗−ab∗Dbxb =(lcxa)(lcxb∗)(degxa−degxb)xdegxa+degxb∗−1+···
8
we have
L L
D b
N (D a)b∗ ab∗ x c =(lc N ) lc ab∗ c (deg a deg b)xmi+1 + .
i(cid:16) x − b (cid:17)ℓY=1 ℓ x i (cid:16) x ℓY=1 ℓ(cid:17) x − x ···
Thiscompletesthe proofthat(Di+1h)/hhasthedenominatorasclaimedandthat
x
its numerator has degree and leading coefficient with respect to x as claimed. The
remaining degree bound with respect to y follows from
L
D c
x ℓ
deg (D N )v iN D v+N v e
y(cid:16) x i − i x i Xℓ=1 ℓ cℓ (cid:17)
L L
∗ ∗
deg c +i deg b +max deg a,deg b + deg c +deg b +deg b+ deg c
≤ y 0 (cid:16) y { y y } Xℓ=1 y ℓ(cid:17) y y Xℓ=1 y ℓ
boundsdegyNi boundsdegyv
| {z } | {z }
L
∗
deg c +(i+1) deg b +max deg a,deg b + deg c
≤ y 0 (cid:16) y { y y } Xℓ=1 y ℓ(cid:17)
and
L
D b
∗ ∗ x
deg N (D a)b ab c
y(cid:16) i(cid:16) x − b (cid:17)ℓY=1 ℓ(cid:17)
L L
∗ ∗
deg c +i deg b +max deg a,deg b + deg c +deg b +deg a+ deg c
≤ y 0 (cid:16) y { y y } Xℓ=1 y ℓ(cid:17) y y Xℓ=1 y ℓ
boundsdegyNi boundsdegy oftheotherfactors
| {z } | {z }
L
∗
deg c +(i+1) deg b +max deg a,deg b + deg c .
≤ y 0 (cid:16) y { y y } Xℓ=1 y ℓ(cid:17)
(2) Again, let v := bb∗ L c and write m for the claimed value of deg N . Then,
ℓ=1 ℓ i x i
like in part 1, Q
L L
(D N )v iN D v+N c (D a)b∗ ab∗Dxb +N v e Dxcℓ
Di+1h x i − i x i ℓ x − b i ℓ cℓ
x = (cid:0)ℓQ=1 (cid:1)(cid:0) (cid:1) ℓP=1 .
h c vi+1
0
First consider the case ω or i ω.
6∈N ≤
Since deg a deg b by assumption, and because of
x ≤ x
(Dxa)b∗−ab∗Dbxb =(lcxa)(lcxb∗)(degxa−degxb)xdegxa+degxb∗−1+··· ,
we now have
L
D b
∗ ∗ x
deg N (D a)b ab c
x(cid:16) i(cid:16) x − b (cid:17)ℓY=1 ℓ(cid:17)
L
∗
<m +deg b+deg b 1+ deg c =m .
i x x − x ℓ i+1
Xℓ=1
9
Note that this estimate is also strict when deg a = deg b because the coefficient
of xdegxa+degxb∗−1 in (Dxa)b∗ −ab∗(Dxb)/b cxontains thxe factor degxa−degxb,
which vanishes in this case.
Next, using the induction hypothesis, we have
L
D c
x ℓ
(D N )v iN D v+N v e
x i i x i ℓ
− c
Xℓ=1 ℓ
L
=(lcxNi)(lcxv)(cid:16)degxNi−idegxv+Xℓ=1eℓdegxcℓ(cid:17)xdegxNi+degxv−1+···
L L
i
=(lcxc0)(cid:16)lcxbb∗ℓY=1cℓ(cid:17) ωi(lcxv)(cid:16)mi−idegxv+Xℓ=1eℓdegxcℓ(cid:17)xmi+degxv−1+···
L L
i+1
=(lc c ) lc bb∗ c ωi deg c + e deg c i xmi+1 +
x 0 (cid:16) x ℓY=1 ℓ(cid:17) (cid:16) x 0 Xℓ=1 ℓ x ℓ− (cid:17) ···
L
i+1
=(lc c ) lc bb∗ c ωi+1xmi+1 + .
x 0 x ℓ
(cid:16) ℓY=1 (cid:17) ···
Since ωi+1 =0 when ω or i+1 ω, this completes the proof that (Di+1h)/h
6 6∈N ≤ x
has the denominator as claimed and that its numerator has degree and leading
coefficient with respect to x as claimed. The degree bounds with respect to y are
shown exactly as in part 1.
Nowconsiderthecasewhereω andi>ω.Inthiscase,westarttheinduction
∈N
ati=ω+1.The inductionbasefollowsfromthecalculationscarriedoutabovefor
i ω, the fact ωω+1 =0, and the definition of δ. (Note that ωω+1 =0 also implies
≥
δ 1.) For the induction step i i+1, we have, similar as before,
≥ 7→
L
D b
∗ ∗ x
deg N (D a)b ab c <m
x(cid:16) i(cid:16) x − b (cid:17)ℓY=1 ℓ(cid:17) i+1
and
L
D c
x ℓ
(D N )v iN D v+N v e
x i i x i ℓ
− c
Xℓ=1 ℓ
L
=(lcxNi)(lcxv)(cid:16)mi−idegxv+Xℓ=1eℓdegxcℓ(cid:17)xmi+degxv−1+···
L
=(lc N )(lc v) deg c +i(deg v 1) δ ideg v+ e deg c xmi+1 +
x i x (cid:16) x 0 x − − − x Xℓ=1 ℓ x ℓ(cid:17) ···
=(lc N ) lc bb∗ L c i−(ω+1)( δ 1)i−(ω+1)(lc v)(ω δ i)xmi+1 +
x ω+1 x ℓ x
(cid:16) ℓY=1 (cid:17) − − − − ···
=(lc N ) lc bb∗ L c i+1−(ω+1)( δ 1)i+1−(ω+1)xmi+1 +
x ω+1 x ℓ
(cid:16) ℓY=1 (cid:17) − − ···
Because of δ >0, the factor ( δ 1)i−(ω+1) is nonzero for all i>ω. ✷
− −
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