Introduction to Gröbner Bases for Geometric Modeling

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 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

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 • 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

5. 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

6. 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

7. 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

8. 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.

9. 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

10. 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

11. 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

12. 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

13. 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

14. 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

15. 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

16. 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

17. 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

18. 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

19. 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...

20. 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

21. 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...

22. 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

23. Gröbner Bases:Buchberger’s Theorem & Construction (continued) Example 7.4 on board... source: Hoffmann

24. Gröbner Bases:Buchberger’s Theorem & Construction (continued) Buchberger’s Theorem: foundation of algorithm Gröbner basis construction algorithm Example 7.5 on board... source: Hoffmann