Introduction to Gr

1
Introduction to Gröbner Bases for Geometric
Modeling

• Geometric Solid Modeling
• 1989
• Christoph M. Hoffmann

2
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
3
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 ?
• Equivalently can g be expressed as an algebraic
  combination of the fj for some polynomials hj?
• 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
4
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
• Buchbergers theorem and construction of Gröbner
  bases
• For discussion of geometric modeling applications
  of Gröbner bases, see Hoffmanns book.
• e.g. Solving simultaneous algebraic expressions
  to find
• surface intersections
• singularities

source Hoffmann
5
Algebraic ConceptsFields, Rings, and Polynomials

• Consider single algebraic equation
• Values of xis are from a field. (Recall from
  earlier in semester.)
• Elements can be added, subtracted, multiplied,
  divided.
• Ground field k is the choice of field .
• Univariate polynomial over k is of form
• Coefficients are numbers in k.
• kx all univariate polynomials using xs.
• 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. x21 factors over complex numbers but not
  real numbers.
• Reducible polynomial can be factored over ground
  field.
• Irreducible polynomial cannot be factored over
  ground field.

for non-0 elements
source Hoffmann
6
Algebraic ConceptsField 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 kx.
• Resulting elements in extended field k(u) are of
  form
• e.g. extending real numbers to complex numbers by
  adjoining i
• i is root of x21, so m2 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 kx.

source Hoffmann
7
Algebraic ConceptsMultivariate Polynomials

• Multivariate polynomial over k is of form
• Coefficients are numbers in k.
• Exponents are nonnegative integers.
• kx1,,xn all multivariate polynomials using
  xs.
• 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.

source Hoffmann
8
Algebraic ConceptsIdeals

• 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 xis
  having values in k.
• f be an irreducible multivariate polynomial in
  kx1,,xn
• g be a multivariate polynomial in kx1,,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
• 
  is an ideal
• g varies over kx1,,xn
• Consider the ideal as the description of the
  surface f.
• Ideal is closed under addition and subtraction.
• Product of an element of kx1,,xn with a
  polynomial in the ideal is in the ideal.

source Hoffmann and others
Ideals are defined more generally in algebra.
9
Algebraic ConceptsIdeals (continued)

• Let F be a finite set of polynomials f1, f2,,
  fr in kx1,,xn
• Algebraic combinations of the fi form an ideal
  generated by F (a generating set)
• generators f, g
• Goal find generating sets, with special
  properties, that are useful for solving geometric
  problems.

Not necessarily unique.
source Hoffmann
10
Algebraic ConceptsAlgebraic Sets

• Let 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 .

source Hoffmann
11
Algebraic ConceptsAlgebraic 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)
• Can define twisted cubic using 3 surfaces
  paraboloid with two cubic surfaces
• Motivation for considering ideals with generating
  sets containing gt 2 polynomials.

source Hoffmann
see Hoffmans Section 7.2.6 for subtleties
related to this statement.
12
Algebraic ConceptsAlgebraic 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 ideals components may have
  different dimensions.

source Hoffmann
13
Algebraic ConceptsAlgebraic Sets and Varieties
(cont.)

• Consider algebraic set V(I) for ideal I in kn.
• V(I) is reducible when V(I) is union of gt 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
14
Algebraic ConceptsAlgebraic Sets and Varieties
(cont.)

• Example Intersection curve of 2 cylinders
• Intersection lies in 2 planes
• and
• Irreducible ellipse in plane
  is is algebraic set in ideal
  generated by f1,g1 .
• Irreducible ellipse in plane is
  is algebraic set in ideal
  generated by f1,g2 .
• Ideal is reducible.
• Decomposes into and
• Algebraic set
• Varieties V(I2) and V(I3)

source Hoffmann
15
Algebraic ConceptsAlgebraic Sets and Varieties
(cont.)

• Example Intersection curve of 2 cylinders
• Intersection curve is not reducible
• These 2 component curves cannot be defined
  separately by polynomials.
• Rationale Bezouts 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.
• Bezouts Theorem also implies that twisted cubic
  cannot be defined algebraically as intersection
  of 2 surfaces
• Twisted cubic has degree 3.
• Bezouts Theorem would imply it is intersection
  of plane and cubic surface.
• But twisted cubic is not planar hence
  contradiction.

Bezouts Theorem 2 irreducible surfaces of
degree m and n intersect in a curve of degree mn.
allowing complex coordinates, points at
infinity
source Hoffmann
16
Gröbner BasesFormulating 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
  where the hi are polynomials.
• Strategy rewrite g until original question can
  be easily answered.

source Hoffmann
17
Gröbner BasesLexicographic Term Ordering and
Leading Terms

• Need to judge if this polynomial is simpler than
  that one.
• Power Product
• Lexicographic ordering of power products
• x
• If then for all power
  products w.
• If u and v are not yet ordered by rules 1 and 2,
  then order them lexicographically as strings.

Example for n2 on board...
source Hoffmann
18
Gröbner BasesLexicographic 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

Example 7.1 on board...
source Hoffmann
19
Gröbner BasesRewriting 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
Example 7.2 on board...
20
Gröbner BasesRewriting 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
21
Gröbner BasesA 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
Example 7.3 on board...
22
Gröbner BasesBuchbergers Theorem Construction

• Algorithm will consist of 2 operations
• Consider a polynomial, and bring it into normal
  form with respect to some set of generators G.
• From certain generator pairs, compute
  S-polynomials (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
23
Gröbner BasesBuchbergers Theorem
Construction (continued)
Example 7.4 on board...
source Hoffmann
24
Gröbner BasesBuchbergers Theorem
Construction (continued)
Buchbergers Theorem foundation of algorithm
Gröbner basis construction algorithm
Example 7.5 on board...
source Hoffmann