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Introduction to Gröbner Bases
for Geometric Modeling
Geometric & Solid Modeling
1989
Christoph M. Hoffmann
Algebraic Geometry
• Branch of mathematics.
• Express geometric facts in algebraic terms in
order to interpret algebraic theorems
geometrically.
• Computations for geometric objects using
symbolic manipulation.
– Surface intersection, finding singularities, and more…
• Historically, methods have been computationally
intensive, so they have been used with
discretion.
source: Hoffmann
Goal
• Operate on geometric object(s) by solving systems of
algebraic equations.
• “Ideal”: (informal partial definition) Set of polynomials
describing a geometric object symbolically.
– Considering algebraic combinations of algebraic equations (without
changing solution) can facilitate solution.
– Ideal is the set of algebraic combinations (to be defined more rigorously later).
– Gröbner basis of an ideal: special set of polynomials defining the
ideal.
• Many algorithmic problems can be solved easily with this basis.
• One example (focus of our lecture): abstract ideal membership problem:
– Is a given polynomial g in a given ideal I ? I  { f1 , f r }
– Equivalently: can g be expressed as an algebraic combination of the fj for
some polynomials hj? g  {h1 f1   hr f r }
– Answer this using Gröbner basis of the ideal.
– Rough geometric interpretation: g can be expressed this way when surface
g = 0 contains all points that are common intersection of surfaces fj = 0.
source: Hoffmann
Overview
• Algebraic Concepts
–
–
–
–
Fields, rings, polynomials
Field extension
Multivariate polynomials and ideals
Algebraic sets and varieties
• Gröbner Bases
–
–
–
–
Lexicographic term ordering and leading terms
Rewriting and normal-form algorithms
Membership test for ideals
Buchberger’s theorem and construction of Gröbner bases
• For discussion of geometric modeling applications of
Gröbner bases, see Hoffmann’s book.
– e.g. Solving simultaneous algebraic expressions to find:
• surface intersections
• singularities
source: Hoffmann
Algebraic Concepts:
Fields, Rings, and Polynomials
• Consider single algebraic equation: f ( x1 ,, xn )  0
• Values of xi’s are from a field. (Recall from earlier in semester.)
– Elements can be added, subtracted, multiplied, divided*.
– Ground field k is the choice of field .
m
* for non-0
elements
• Univariate polynomial over k is of form:  ai x i
i 0
– Coefficients are numbers in k.
– k[x] = all univariate polynomials using x’s.
• It is a ring (recall from earlier in semester): addition, subtraction,
multiplication, but not necessarily division.
• Can a given polynomial be factored?
– Depends on ground field
• e.g. x2+1 factors over complex numbers but not real numbers.
– Reducible: polynomial can be factored over ground field.
– Irreducible: polynomial cannot be factored over ground field.
source: Hoffmann
Algebraic Concepts:
Field Extension
• Field extension: enlarging a field by adjoining
(adding) new element(s) to it.
– Algebraic Extension:
• Adjoin an element u that is a root of a polynomial (of degree m) in k[x].
– Resulting elements in extended field k(u) are of form:
a0  a1u  a2u 2   am1u m1
– e.g. extending real numbers to complex numbers by adjoining i
» i is root of x2+1, so m=2 and extended field elements are of
form a + bi
– e.g. extending rational numbers to algebraic numbers by adjoining
roots of all univariate polynomials (with rational coefficients)
– Transcendental Extension:
• Adjoin an element (such as p) that is not the root of any polynomial in
k[x].
source: Hoffmann
Algebraic Concepts:
Multivariate Polynomials
m
• Multivariate polynomial over k is of form:  a j x1e1, j x2 e2 , j xn en , j
j 1
– Coefficients are numbers in k.
– Exponents are nonnegative integers.
– k[x1,…,xn] = all multivariate polynomials using x’s.
• It is a ring: addition, subtraction, multiplication, but not
necessarily division.
• Can a given polynomial be factored?
– Depends on ground field (as in univariate case)
– Reducible: polynomial can be factored over ground field.
– Irreducible: polynomial cannot be factored over ground field.
– Absolutely Irreducible: polynomial cannot be factored over any
ground field.
• e.g. x 2  y 2  z 2  1
source: Hoffmann
Algebraic Concepts:
Ideals
• For ground field k, let:
– kn be the n-dimensional affine space over k.
• mathematical physicist John Baez: "An affine space is a vector space
that's forgotten its origin”.
– Points in kn are n-tuples (x1,…,xn), with xi’s having values in k.
– f be an irreducible multivariate polynomial in k[x1,…,xn]
– g be a multivariate polynomial in k[x1,…,xn]
– f = 0 be the hypersurface in kn defined by f
• Since hypersurface gf = 0 includes f = 0, view f as intersection of all
surfaces of form gf = 0
•
I f  {gf  k[ x1 ,..., xn ] | f fixed} is an ideal*
–
–
–
–
g varies over k[x1,…,xn]
Consider the ideal as the description of the surface f.
Ideal is closed under addition and subtraction.
Product of an element of k[x1,…,xn] with a polynomial in the ideal is
in the ideal.
*Ideals are defined more generally in algebra.
source: Hoffmann and others
Algebraic Concepts:
Ideals (continued)
• Let F be a finite set of polynomials f1, f2,…, fr in
k[x1,…,xn]
• Algebraic combinations of the fi form an ideal
generated by F (a generating set*):
I F  {g1 f1  g2 f 2  ...  gr f r | gi  k[ x1 ,..., xn ]}
– generators: { f, g }
• Goal: find generating sets, with special
properties, that are useful for solving geometric
problems.
* Not necessarily unique.
source: Hoffmann
Algebraic Concepts:
Algebraic Sets
• Let I  k[ x1 ,..., xn ] be the ideal generated by the
finite set of polynomials F = { f1, f2,…, fr }.
• Let p = (a1,…, an) be a point in kn such that g(p) =
0 for every g in I.
• Set of all such points p is the algebraic set V(I)
of I.
– It is sufficient that fi(p) = 0 for every generator fi in F.
• In 3D, the algebraic surface f = 0 is the
algebraic set of the ideal I f .
source: Hoffmann
Algebraic Concepts:
Algebraic Sets (cont.)
•
Intersection of two algebraic surfaces f, g in
3D is an algebraic space curve.
– The curve is the algebraic set of the ideal.
•
•
But, not every algebraic space curve can be
defined as the intersection of 2 surfaces.
Example where 3 are needed*: twisted
cubic (in parametric form):
xt
y  t2
z  t3
•
Can define twisted cubic using 3 surfaces:
paraboloid with two cubic surfaces:
x
•
2
 
 
 y  0  y 3  z 2  0  z  x3

Motivation for considering ideals with
generating sets containing > 2 polynomials.
*see Hoffman’s Section 7.2.6 for subtleties related to this statement.
source: Hoffmann
Algebraic Concepts:
Algebraic Sets and Varieties (cont.)
• Given generators F = { f1, f2,…, fr }, the
algebraic set defined by F in kn has
dimension n-r
– If equations fi = 0 are algebraically
independent.
– Complication: some of ideal’s components
may have different dimensions.
source: Hoffmann
Algebraic Concepts:
Algebraic Sets and Varieties (cont.)
• Consider algebraic set V(I) for ideal I in kn.
• V(I) is reducible when V(I) is union of > 2
point sets, each defined separately by an
ideal.
– Analogous to polynomial factorization:
• Multivariate polynomial f that factors describes
surface consisting of several components
– Each component is an irreducible factor of f.
• V(I) is irreducible implies V(I) is a variety.
source: Hoffmann
Algebraic Concepts:
Algebraic Sets and Varieties (cont.)
• Example: Intersection curve of 2
cylinders: ff :: xy  yz  rr  00
• Intersection lies in 2 planes:
g1 : x  z  0 and g2 : x  z  0
• Irreducible ellipse in plane g1 : x  z  0 is
is algebraic set in ideal I 2  I f1 , g1
generated by { f1,g1 }.
• Irreducible ellipse in plane g2 : x  z  0 is is
algebraic set in ideal I 3  I f1 , g2
generated by { f1,g2 }.
• Ideal I1  I f1, f 2 is reducible.
2
2
2
2
2
2
1
2
– Decomposes into I 2  I f1 , g1
and
I 3  I f1 , g 2
• Algebraic set V ( I1 )  V ( I 2 ) V ( I 3 )
– Varieties: V(I2) and V(I3)
source: Hoffmann
Algebraic Concepts:
Algebraic Sets and Varieties (cont.)
•
Example: Intersection curve of 2 cylinders:
f1 : x 2  y 2  1  0
f2 : y2  z 2  2  0
– Intersection curve is not reducible
• These 2 component curves cannot be defined
separately by polynomials.
• Rationale: Bezout’s Theorem implies
intersection curve has degree 4. Furthermore:
– Union of 2 curves of degree m and n is a
reducible curve of degree m + n.
– If intersection curve were reducible, then
consider degree combinations for component
curves (total = 4):
» 1 + 3: illegal since neither has degree 1.
» 2 + 2: illegal since neither is planar.
» Conclusion: intersection curve irreducible.
•
Bezout’s Theorem also implies that twisted
cubic cannot be defined algebraically as
intersection of 2 surfaces:
• Twisted cubic has degree 3.
• Bezout’s Theorem would imply it is intersection
of plane and cubic surface.
• But twisted cubic is not planar; hence
contradiction.
Bezout’s Theorem*: 2 irreducible surfaces of degree m and n intersect in a curve of
source: Hoffmann
degree mn.
*allowing complex coordinates, points at infinity
Gröbner Bases:
Formulating Ideal Membership Problem
• Can help to solve geometric modeling problems
such as intersection of implicit surfaces (see
Hoffmann Sections 7.4-7.8).
• Here we only treat the ideal membership
problem for illustrative purposes:
– “Given a finite set of polynomials F = { f1, f2,…, fr },
and a polynomial g, decide whether g is in the ideal
generated by F; that is, whether g can be written in
the form g  h1 f1  h2 f 2  ...  hr f r where the hi are
polynomials.”
• Strategy: rewrite g until original question can be
easily answered.
source: Hoffmann
Gröbner Bases:
Lexicographic Term Ordering and Leading
Terms
•
•
•
Need to judge if “this polynomial is simpler
than that one.”
Power Product: x x  x , e  0
Lexicographic ordering of power products:
e1
e2
1
2
en
n
i
1. 1x x1  x2    xn
2. If u  v then uw  vw for all power products w.
3. If u and v are not yet ordered by rules 1 and 2, then
order them lexicographically as strings.
source: Hoffmann
Gröbner Bases:
Lexicographic Term Ordering and Leading
Terms
•
Each term in a polynomial g is a coefficient
combined with a power product.
– Leading term lt(g) of g: term whose power product
is largest with respect to ordering 
•
•
•
lcf (g) =leading coefficient of lt(g)
lpp (g) =leading power product of lt(g)
Definition: Polynomial f is simpler than
polynomial g if: lpp ( f )  lpp ( g )
source: Hoffmann
Gröbner Bases:
Rewriting and Normal-Form Algorithms
•
Given polynomial g and set of polynomials F = { f1, f2,…, fr }
– Rewrite/simplify g using polynomials in F.
– g is in normal form NF(g, F) if it cannot be reduced
further. Note: normal form need not be unique.
source: Hoffmann
Gröbner Bases:
Rewriting and Normal-Form Algorithms
•
If normal form from rewriting algorithm is unique
– then g is in ideal when NF(g, F) = 0.
•
This motivates search for generating sets that
produce unique normal forms.
source: Hoffmann
Gröbner Bases:
A Membership Test for Ideals
•
Goal: Rewrite g to decide whether g is in the ideal
generated by F.
– Gröbner basis G of ideal
•
Set of polynomials generating F.
•
Rewriting algorithm using G produces unique normal forms.
– Ideal membership algorithm using G:
source: Hoffmann
Gröbner Bases:
Buchberger’s Theorem & Construction
•
Algorithm will consist of 2 operations:
1. Consider a polynomial, and bring it into normal form
with respect to some set of generators G.
2. From certain generator pairs, compute Spolynomials (see definition on next slide) and add their
normal forms to the generator set.
•
•
G starts as input set F of polynomials
G is transformed into a Gröbner basis.
•
Some Implementation Issues:
–
Coefficient arithmetic must be exact.
•
–
Rational arithmetic can be used.
Size of generator set can be large.
•
Reduced Gröbner bases can be developed.
source: Hoffmann
Gröbner Bases:
Buchberger’s Theorem & Construction (continued)
source: Hoffmann
Gröbner Bases:
Buchberger’s Theorem & Construction (continued)
Buchberger’s
Theorem:
foundation of
algorithm
Gröbner basis
construction
algorithm
source: Hoffmann