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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 ?
- Equivalently: can g be expressed as an algebraic combination of the fjfor 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

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:
- Values of xi’s are from a field.(Recall from earlier in semester.)
- Elements can be added, subtracted, multiplied, divided*.
- Ground fieldk is the choice of field .
- Univariate polynomial over k is of form:
- 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.

* for non-0 elements

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:
- 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

- Multivariate polynomial over k is of form:
- 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.

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]
- gbe 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
- 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.

source: Hoffmann and others

*Ideals are defined more generally in algebra.

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*):
- 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 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 setV(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

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):
- Can define twisted cubic using 3 surfaces: paraboloid with two cubic surfaces:
- 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 generatorsF = { 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:
- 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

Algebraic Concepts:Algebraic 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: 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 degree mn. *allowing complex coordinates, points at infinity

source: Hoffmann

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 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:
- 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 n=2 on board...

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:

Example 7.1 on board...

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.
- gis in normal formNF(g, F) if it cannot be reduced further. Note: normal form need not be unique.

source: Hoffmann

Example 7.2 on board...

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 basisG of ideal
- Set of polynomials generating F.
- Rewriting algorithm using G produces unique normal forms.
- Ideal membership algorithm usingG:

source: Hoffmann

Example 7.3 on board...

Gröbner Bases:Buchberger’s 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

Gröbner Bases:Buchberger’s Theorem & Construction (continued)

Example 7.4 on board...

source: Hoffmann

Gröbner Bases:Buchberger’s Theorem & Construction (continued)

Buchberger’s Theorem:

foundation of algorithm

Gröbner basis construction algorithm

Example 7.5 on board...

source: Hoffmann

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