Fundamental problems in algorithmic algebra

550 Pages·2000·4.32 MB·English

by  Yap C.K.

Fundamental Problems in Algorithmic Algebra
Chee Keng Yap
Courant Institute of Mathematical Sciences
New York University
251 Mercer Street
New York, NY 10012
September 8, 1993
TO BE PUBLISHED BY PRINCETON UNIVERSITY PRESS
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requested to inform the author of any errors, typographical or otherwise. All suggestions welcome. Electronic
mail: [email protected].
(cid:13)c Chee-Keng Yap September 8, 1993
Contents
0. Introduction
I. Basic Arithmetic
II. The GCD
III. Subresultants
IV. Modular Techniques: Chinese Remainder
V. Fundamental Theorem of Algebra
VI. Roots of Polynomials
VII. Sturm Theory
VIII. Gaussian Lattice Reduction
IX. Lattices and Polynomial Factorization
X. Elimination Theory
XI. Gr¨obner Bases
XII. Continued Fractions
(cid:13)c Chee-Keng Yap September 8, 1993
PREFACE
These notes were first written for a course on Algebraic Computing: Solving Systems of Poly-
nomial Equations, given in the Spring Semester of 1989 at the Free University of Berlin. They
werethoroughlyrevisedfollowingasimilarcourseattheCourantInstituteintheSpringof1992.
Prerequisites are an undergraduate course in algebra and a graduate course in algorithmics.
I regard this course as an introduction to computer algebra. The subject matter (‘starting
from the Fundamental Theorem of Algebra’) is as classical as one gets in theoretical computer
science, and yet it is refreshingly contemporary in interest. This is because the complexity
viewpoint exposes many classicalquestions to new light. There is a common misunderstanding
that equates computational mathematics with numerical analysis. In fact, it seems to me that
the older name of “symbolic manipulation” given to our field arose as a direct contrast to
“numerical computation”. The preferred name today is “computer algebra”, although I feel
that “algorithmic algebra” gives a better emphasis to the fundamental nature of the subject.
Inanycase,computeralgebrausesquite distincttechniques,andsatisfiesrequirementsdistinct
from that in numerical analysis. In many areas of computer application (robotics, computer-
aided design, geometricmodeling, etc) computer algebrais now recognizedas anessentialtool.
This is partly driven by the wide-spread availability of powerful computer work-stations, and
the rise of a new generation of computer algebra systems to take advantage of this computing
power.
The full spectrum of activity in computer algebra today covers many important areas that
we do not even give a hint of in these lectures: it ranges from more specialized topics such
as algorithmic integration theory, to implementation issues in computer algebra systems, to a
highly developed and beautiful complexity theory of algebraic problems, to problems in allied
applicationareassuchasrobotmotionplanning. Ourmaterialisnecessarilyselective,although
we feel that if one must cut one swathfrom the elementaryinto the deeper parts of the subject
inanintroductorycourse,this isachoicecut. Historically,whatweidentifiedas“Fundamental
problems” in these lectures were clearly central to the development of algebra and even of
mathematics. Thereisanenormousamountofrelevantclassicalliteratureonthesefundamental
problems, in part a testimony to the strong algorithmic nature of mathematics before the
twentieth century. Even when restricted to this corpus of knowledge (classical, supplemented
by modern algorithmic development), my colleagues will surely notice important gaps. But I
hope they may still find this book useful as a launching point into their own favorite areas.
We have tried to keep the style of the book close to the lecture form in which this material
originally existed. Of course, we have considerably expanded on the lecture material. This
mainly consisted of the filling in of mathematical background: a well-equipped student may
skip this. The teacher could convey the central ideas quickly at the expense of generality, for
instance, by assuming that the rings under discussion are the “canonical examples” ( and
Z
F[X]). One teaching plan is to choose a subset of the material in each Lecture Section of this
book for presentation in a 2-hour class (the typical length of class at Courant), with the rest
assigned for further reading.
I thank Frau Schottke from the Free University for her dedicated transcription of my original
hand-written notes into the computer.
(cid:13)c Chee-Keng Yap September 8, 1993
Chee Yap
Greenwich Village
September 8, 1993
(cid:13)c Chee-Keng Yap September 8, 1993
§1. Problem of Algebra Lecture 0 Page 1
Lecture 0
INTRODUCTION
Thislectureisanorientationonthe centralproblemsthatconcernus. Specifically,weidentifythree
families of “Fundamental Problems”in algorithmic algebra( 1 – 3). In the restof the lecture ( 4–
§ § §
9),webrieflydiscussthecomplexity-theoreticbackground. 10collectssomecommonmathematical
§ §
terminology while 11 introduces computer algebra systems. The reader may prefer to skip 4-11
§ §
on a first reading, and only use them as a reference.
All our rings will contain unity which is denoted 1 (and distinct from 0). They
are commutative except in the case of matrix rings.
The main algebraic structures of interest are:
N = natural numbers 0,1,2,...
Z = integers
Q = rational numbers
R = reals
C = complex numbers
R[X] = polynomial ring in d 1 variables X=(X ,...,X )
1 n
≥
with coefficients from a ring R.
Let R be any ring. For a univariate polynomial P R[X], we let deg(P) and lead(P) denote its
∈
degreeandleading coefficient(orleadingcoefficient). IfP =0thenbydefinition,deg(P)= and
−∞
lead(P)=0;otherwisedeg(P) 0andlead(P)=0. WesayP isa(respectively)integer,rational,
≥ 6
real or complex polynomial, depending on whether R is Z,Q,R or C.
In the course of this book, we will encounter other rings: (e.g., I.1). With the exception of matrix
§
rings, all our rings are commutative. The basic algebra we assume can be obtained from classics
such as van der Waerden [22] or Zariski-Samuel[27, 28].
§1. Fundamental Problem of Algebra
Consider an integer polynomial
n
P(X)= a Xi (a Z,a =0). (1)
i i n
∈ 6
i=0
X
Many of the oldest problems in mathematics stem from attempts to solve the equation
P(X)=0, (2)
i.e., to find numbers α such that P(α) = 0. We call such an α a solution of equation (2); alterna-
tively,αisarootorzeroofthepolynomialP(X). Bydefinition,analgebraic numberisazeroofsome
polynomial P Z[X]. The Fundamental Theorem of Algebra states that every non-constant poly-
∈
nomial P(X) C[X]has a root α C. Put another way,C is algebraicallyclosed. d’Alembert first
∈ ∈
formulated this theorem in 1746 but Gauss gave the first complete proof in his 1799 doctoral thesis
(cid:13)c Chee-Keng Yap March 6, 2000
§1. Problem of Algebra Lecture 0 Page 2
atHelmstedt. Itfollowsthattherearen(notnecessarilydistinct)complexnumbersα ,...,α C
1 n
∈
such that the polynomial in (1) is equal to
n
P(X) a (X α ). (3)
n i
≡ −
i=1
Y
To see this, suppose α is a root of P(X) as guaranteed by the Fundamental Theorem. Using the
1
synthetic division algorithm to divide P(X) by X α , we get
1
−
P(X)=Q (X) (X α )+β
1 1 1
· −
where Q (X) is a polynomial of degree n 1 with coefficients in C and β C. On substituting
1 1
− ∈
X = α , the left-hand side vanishes and the right-hand side becomes β . Hence β = 0. If n = 1,
1 1 1
then Q (X) = a and we are done. Otherwise, this argument can be repeated on Q (X) to yield
1 n 1
equation (3).
The computational version of the Fundamental Theorem of Algebra is the problem of finding roots
of a univariate polynomial. We may dub this the Fundamental Problem of Computational Algebra
(or Fundamental Computational Problem of Algebra). The FundamentalTheoremis about complex
numbers. For our purposes, we slightly extend the context as follows. If R R are rings, the
0 1
⊆
Fundamental Problem for the pair (R ,R ) is this:
0 1
Given P(X) R [X], solve the equation P(X)=0 in R .
0 1
∈
We are mainly interested in cases where Z R R C. The three main versions are where
0 1
⊆ ⊆ ⊆
(R ,R ) equals (Z,Z),(Z,R) and (Z,C), respectively. We call them the Diophantine, real and
0 1
complex versions (respectively) of the Fundamental Problem.
What does it mean “to solveP(X)=0 in R ”? The most naturalinterpretationis that we wantto
1
enumeratealltherootsofP thatlieinR . Besidesthisenumeration interpretation,weconsidertwo
1
other possibilities: the existential interpretation simply wants to know if P has a root in R , and
1
the counting interpretation wants to know the number of such roots. To enumerate1 roots,we must
addressthe representationof these roots. For instance,we will study a representationvia “isolating
intervals”.
Recall another classical version of the Fundamental Problem. Let R = Z and R denote the
0 1
complex subring comprising all those elements that can be obtained by applying a finite number of
field operations (ring operations plus division by non-zero) and taking nth roots (n 2), starting
≥
fromZ. Thisisthefamoussolutionbyradicals versionoftheFundamentalProblem. Itiswellknown
thatwhendegP =2,thereisalwaysasolutioninR . WhatifdegP >2? Thiswasamajorquestion
1
of the 16th century, challenging the best mathematicians of its day. We now know that solution
by radicals exists for degP =3 (Tartaglia, 1499-1557)and degP =4 (variously ascribed to Ferrari
(1522-1565)or Bombelli (1579)). These methods were widely discussed, especially after they were
published by Cardan (1501-1576) in his classic Ars magna, “The Great Art”, (1545). This was the
algebra book until Descartes’ (1637) and Euler’s Algebra (1770). Abel (1824) (also Wantzel) show
thatthereis nosolutionbyradicalsfora generalpolynomialofdegree5. Ruffinihadapriorthough
incomplete proof. This kills the hope fora single formula whichsolvesall quintic polynomials. This
still leaves open the possibility that for each quintic polynomial, there is a formula to extract its
roots. But itis nothardto dismiss this possibility: forexample, anexplicit quintic polynomialthat
1Thereispossibleconfusionhere: theword“enumerate”meansto“count”aswellasto“listbyname”. Sincewe
areinterestedinbothmeaningshere,wehavetoappropriatetheword“enumerate”foronlyoneofthesetwosenses.
Inthisbook,wetrytouseitonlyinthelattersense.
(cid:13)c Chee-Keng Yap March 6, 2000
§2. Algebraic Geometry Lecture 0 Page 3
does not admit solution by radicals is P(X) = X5 16X +2 (see [3, p.574]). Miller and Landau
−
[12] (also[26]) revisitsthese questionfroma complexity viewpoint. The abovehistoricalcomments
may be pursued more fully in, for example, Struik’s volume [21].
Remarks:. The Fundamental Problem of algebra used to come under the rubric “theory of equa-
tions”, which nowadays is absorbed into other areas of mathematics. In these lectures, we are
interested in general and effective methods, and we are mainly interested in real solutions.
§2. Fundamental Problem of Classical Algebraic Geometry
To generalizethe Fundamental Problemof algebra,we continue to fix two rings,Z R R C.
0 1
⊆ ⊆ ⊆
First consider a bivariate polynomial
P(X,Y) R [X,Y]. (4)
0
∈
Let Zero(P) denote the set of R -solutions of the equation P = 0, i.e., (α,β) R2 such that
P(α,β) = 0. The zero set Zero(1P) of P is generally an infinite set. In case R∈ =1R, the set
1
Zero(P) is a planar curve that can be plotted and visualized. Just as solutions to equation (2) are
called algebraic numbers, the zero sets of bivariate integer polynomials are called algebraic curves.
But there is no reason to stop at two variables. For d 3 variables, the zero set of an integer
≥
polynomial in d variables is called an algebraic hypersurface: we reserve the term surface for the
special case d=3.
Given two surfaces defined by the equations P(X,Y,Z) = 0 and Q(X,Y,Z)= 0, their intersection
is generally a curvilinear set of triples (α,β,γ) R3, consisting of all simultaneous solutions to the
∈ 1
pair of simultaneous equations P = 0, Q = 0. We may extend our previous notation and write
Zero(P,Q) for this intersection. Moregenerally,we wantthe simultaneous solutions to a system of
m 1 polynomial equations in d 1 variables:
≥ ≥
P =0
1
P =0
2
...  (where Pi ∈R0[X1,...,Xd]) (5)
P =0
m
A point (α ,...,α ) Rd is called a solution of the system of equations (5) or a zero of the set
1 d ∈ 1
P ,...,P provided P (α ,...,α ) = 0 for i = 1,...,m. In general, for any subset J R [X],
1 m i 1 d 0
{let Zero(J)} Rd denote the zero set of J. To denote the dependence on R , we may a⊆lso write
Zero (J). ⊆If R1 is a field, we also call a zero set an algebraic set. Since1the primary objects
R1 1
of study in classical algebraic geometry are algebraic sets, we may call the problem of solving the
system (5) the Fundamental (Computational) Problem of classical algebraic geometry. If each P is
i
linear in (5), we arelooking ata systemof linear equations. One mightcallthis is the Fundamental
(Computational) Problem of linear algebra. Ofcourse,linearsystems arewellunderstood,andtheir
solution technique will form the basis for solving nonlinear systems.
Again,wehavethreenaturalmeaningstotheexpression“solvingthesystemofequations(5)inR ”:
1
(i)TheexistentialinterpretationasksifZero(P ,...,P )isempty. (ii)Thecountinginterpretation
1 m
asks for the cardinality of the zero set. In case the cardinality is “infinity”, we could refine the
question by asking for the dimension of the zero set. (iii) Finally, the enumeration interpretation
poses no problems when there are only finitely many solutions. This is because the coordinates of
these solutions turn out to be algebraic numbers and so they could be explicitly enumerated. It
becomes problematic when the zero set is infinite. Luckily, when R = R or C, such zero sets are
1
well-behaved topologically, and each zero set consists of a finite number of connected components.
(cid:13)c Chee-Keng Yap March 6, 2000
§3. Ideal Theory Lecture 0 Page 4
(For that matter, the counting interpretation can be re-interpreted to mean counting the number
of components of each dimension.) A typical interpretation of “enumeration” is “give at least one
sample point from each connected component”. For real planar curves, this interpretation is useful
for plotting the curve since the usual method is to “trace” each component by starting from any
point in the component.
Note thatwe havemovedfromalgebra(numbers)to geometry(curvesandsurfaces). Inrecognition
ofthis,weadoptthegeometriclanguageof“pointsandspace”. ThesetRd(d-foldCartesianproduct
1
ofR )iscalledthed-dimensional affine spaceofR ,denotedAd(R ). ElementsofAd(R )arecalled
1 1 1 1
d-points or simply points. Our zero sets are subsets of this affine space Ad(R ). In fact, Ad(R ) can
1 1
be given a topology (the Zariski topology) in which zero sets are the closed sets.
There are classical techniques via elimination theory for solving these Fundamental Problems. The
recent years has seen a revival of these techniques as well as major advances. In one line of work,
Wu Wen-tsun exploited Ritt’s idea of characteristic sets to give new methods for solving (5) rather
efficiently in the complex case, R =C. These methods turn out to be useful for proving theorems
1
in elementary geometry as well [25]. But many applications are confined to the real case (R =R).
1
Unfortunately, it is a general phenomenon that real algebraic sets do not behave as regularly as
the corresponding complex ones. This is already evident in the univariate case: the Fundamental
Theorem of Algebra fails for realsolutions. In view of this, most mathematical literature treats the
complex case. More generally, they apply to any algebraically closed field. There is now a growing
body of results for real algebraic sets.
Another step traditionally taken to “regularize” algebraic sets is to consider projective sets, which
abolishthe distinction between finite andinfinite points. A projective d-dimensional point is simply
anequivalenceclassofthesetAd+1(R ) (0,...,0) ,wheretwonon-zero(d+1)-pointsareequivalent
1
\{ }
if one is a constant multiple of the other. We use Pd(R ) to denote the d-dimensional projective
1
space of R .
1
Semialgebraic sets. The realcase admits a generalizationof the system(5). We canview (5) as
a conjunction of basic predicates of the form “P =0”:
i
(P =0) (P =0) (P =0).
1 2 m
∧ ∧···∧
We generalizethis to an arbitraryBooleancombination of basic predicates,where a basic predicate
now has the form (P =0) or (P >0) or (P 0). For instance,
≥
((P =0) (Q>0)) (R 0)
∧ ∨ ¬ ≥
is a Boolean combination of three basic predicates where P,Q,R are polynomials. The set of real
solutions to such a predicate is called a semi-algebraic set (or a Tarski set). We have effective
methods of computing semi-algebraic sets, thanks to the pioneering work of Tarski and Collins [7].
Recent work by various researchers have reduced the complexity of these algorithms from double
exponentialtime tosingleexponentialspace[15]. This surveyalsodescribestoapplicationsofsemi-
algebraic in algorithmic robotics, solid modeling and geometric theorem proving. Recent books on
real algebraic sets include [4, 2, 10].
§3. Fundamental Problem of Ideal Theory
Algebraic sets are basically geometric objects: witness the language of “space, points, curves, sur-
faces”. Nowweswitchfromthegeometricviewpoint(back!) toanalgebraicone. Oneofthebeauties
of this subject is this interplay between geometry and algebra.
(cid:13)c Chee-Keng Yap March 6, 2000
§3. Ideal Theory Lecture 0 Page 5
Fix Z R R C as before. A polynomial P(X) R [X] is said to vanish on a subset
0 1 0
⊆ ⊆ ⊆ ∈
U Ad(R ) if for all a U, P(a)=0. Define
1
⊆ ∈
Ideal(U) R [X]
0
⊆
to comprise allpolynomials P R [X] that vanishon U. The set Ideal(U) is an ideal. Recallthat
0
∈
a non-empty subset J R of a ring R is an ideal if it satisfies the properties
⊆
1. a,b J a b J
∈ ⇒ − ∈
2. c R, a J ca J.
∈ ∈ ⇒ ∈
For any a1,...,am R and R′ R, the set (a1,...,am)R′ defined by
∈ ⊇
m
(a1,...,am)R′:= aibi :b1,...,bm R′
{ ∈ }
i=1
X
is an ideal, the ideal generated by a ,...,a in R′. We usually omit the subscript R′ if this is
1 m
understood.
The FundamentalProblemofclassicalalgebraicgeometry(see Equation(5))canbe viewedascom-
puting (some characteristic property of) the zero set defined by the input polynomials P ,...,P .
1 m
But note that
Zero(P ,...,P )=Zero(I)
1 m
where I is the ideal generated by P ,...,P . Hence we might as well assume that the input to the
1 m
FundamentalProblemis the idealI (representedby a setofgenerators). This suggests that we view
ideals to be the algebraic analogue of zero sets. We may then ask for the algebraic analogue of the
FundamentalProblemofclassicalalgebraicgeometry. Anaive answeris that, “givenP ,...,P , to
1 m
enumerate the set (P ,...,P )”. Of course, this is impossible. But we effectively “know” a set S
1 m
if, for any purported member x, we can decisively say whether or not x is a member of S. Thus we
reformulate the enumerative problem as the Ideal Membership Problem:
Given P ,P ,...,P R [X], is P in (P ,...,P )?
0 1 m 0 0 1 m
∈
WheredoesR comein? Well,theideal(P ,...,P )isassumedtobegeneratedinR [X]. Weshall
1 1 m 1
introduce effective methods to solve this problem. The technique of Gr¨obner bases (as popularized
by Buchberger) is notable. There is strong historical basis for our claim that the ideal membership
problem is fundamental: van der Waerden [22, vol. 2, p. 159] calls it the “main problem of ideal
theory in polynomial rings”. Macaulay in the introduction to his 1916 monograph [14] states that
the “object of the algebraic theory [of ideals] is to discover those general properties of [an ideal]
which will afford a means of answering the question whether a given polynomial is a member of a
given [ideal] or not”.
How generalarethe ideals ofthe form(P ,...,P )? The onlyideals that mightnotbe ofthis form
1 m
are those that cannot be generated by a finite number of polynomials. The answer is provided by
what is perhaps the starting point of modern algebraic geometry: the Hilbert!Basis Theore. A ring
R is called Noetherian if all its ideals are finitely generated. For example, if R is a field, then it
is Noetherian since its only ideals are (0) and (1). The Hilbert Basis Theorem says that R[X] is
Noetherianif R is Noetherian. This theoremis crucial2 froma constructiveviewpoint: it assuresus
that although ideals are potentially infinite sets, they are finitely describable.
2The paradox is, many view the original proof of this theorem as initiating the modern tendencies toward non-
constructiveproofmethods.
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§3. Ideal Theory Lecture 0 Page 6
We now have a mapping
U Ideal(U) (6)
7→
from subsets of Ad(R ) to the ideals of R [X], and conversely a mapping
1 0
J Zero(J) (7)
7→
from subsets of R [X] to algebraic sets of Ad(R ). It is not hard to see that
0 1
J Ideal(Zero(J)), U Zero(Ideal(U)) (8)
⊆ ⊆
for all subsets J R [X] and U Ad(R ). Two other basic identities are:
0 1
⊆ ⊆
Zero(Ideal(Zero(J))) = Zero(J), J R [X],
0
⊆
Ideal(Zero(Ideal(U))) = Ideal(U), U Ad(R ), (9)
1
⊆
We prove the first equality: If a Zero(J) then for all P Ideal(Zero(J)), P(a) = 0. Hence
a Zero(Ideal(Zero(J)). Co∈nversely, if a Zero(Ide∈al(Zero(J)) then P(a) = 0 for all
P ∈ Ideal(Zero(J)). But since J Ideal(Ze∈ro(J)), this means that P(a) = 0 for all P J.
He∈nce a Zero(J). The second equ⊆ality (9) is left as an exercise. ∈
∈
If we restrict the domain of the map in (6) to algebraic sets and the domain of the map in (7)
to ideals, would these two maps be inverses of each other? The answer is no, based on a simple
observation: AnidealI iscalledradical ifforallintegersn 1,Pn I impliesP I. Itisnothard
to check that Ideal(U) is radical. On the other hand, the≥ideal (X∈2) Z[X] is cl∈early non-radical.
∈
It turns out that if we restrict the ideals to radical ideals, then Ideal() and Zero() would be
· ·
inverses of each other. This is captured in the Hilbert Nullstellensatz (or, Hilbert’s Zero Theorem
in English). After the Basis Theorem, this is perhaps the next fundamental theorem of algebraic
geometry. ItstatesthatifP vanishesonthe zerosetofanidealI thensomepowerPn ofP belongs
to I. As a consequence,
I =Ideal(Zero(I)) I is radical.
⇔
Inproof: Clearlythe left-handside impliesI isradical. Conversely,ifI is radical,itsuffices toshow
that Ideal(Zero(I)) I. Say P Ideal(Zero(I)). Then the Nullstellensatz implies Pn I for
⊆ ∈ ∈
some n. Hence P I since I is radical, completing our proof.
∈
We nowhaveabijective correspondencebetweenalgebraicsetsandradicalideals. This implies that
ideals in general carry more information than algebraic sets. For instance, the ideals (X) and (X2)
have the same zero set, viz., X =0. But the unique zero of (X2) has multiplicity 2.
Theideal-theoreticapproach(oftenattachedtothenameofE.Noether)characterizesthetransition
from classical to “modern” algebraic geometry. “Post-modern” algebraic geometry has gone on to
more abstract objects such as schemes. Not much constructive questions are raised at this level,
perhaps because the abstract questions are hard enough. The reader interested in the profound
transformation that algebraic geometry has undergone over the centuries may consult Dieudonn´e
[9] who described the subject in “seven epochs”. The current challenge for constructive algebraic
geometry appears to be at the levels of classicalalgebraic geometry and at the ideal-theoretic level.
For instance, Brownawell [6]and others have recently given us effective versions of classical results
such as the Hilbert Nullstellensatz. Such results yields complexity bounds that are necessary for
efficient algorithms (see Exercise).
This concludes our orientation to the central problems that motivates this book. This exercise is
pedagogically useful for simplifying the algebraic-geometric landscape for students. However, the
richness of this subject and its complex historical development ensures that, in the opinion of some
(cid:13)c Chee-Keng Yap March 6, 2000