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Introduction to Gröbner Bases for Geometric Modeling Geometric & Solid Modeling 1989 Christoph M. Hoffmann

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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

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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 f j for some polynomials h j ? –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 f j = 0. source: Hoffmann

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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

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Algebraic Concepts: Fields, Rings, and Polynomials Consider single algebraic equation: Values of x i ’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. 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. x 2 +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 * for non-0 elements

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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 x 2 +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 ) that is not the root of any polynomial in k[x]. source: Hoffmann

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Algebraic Concepts: Multivariate Polynomials Multivariate polynomial over k is of form: –Coefficients are numbers in k. –Exponents are nonnegative integers. – k[x 1,…,x n ] = 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

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Algebraic Concepts: Ideals For ground field k, let: – k n 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 k n are n -tuples ( x 1,…,x n ), with x i ’s having values in k. –f be an irreducible multivariate polynomial in k[x 1,…,x n ] –g be a multivariate polynomial in k[x 1,…,x n ] –f = 0 be the hypersurface in k n 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[x 1,…,x n ] –Consider the ideal as the description of the surface f. –Ideal is closed under addition and subtraction. –Product of an element of k[x 1,…,x n ] with a polynomial in the ideal is in the ideal. source: Hoffmann and others *Ideals are defined more generally in algebra.

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Algebraic Concepts: Ideals (continued) Let F be a finite set of polynomials f 1, f 2,…, f r in k[x 1,…,x n ] Algebraic combinations of the f i 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. source: Hoffmann * Not necessarily unique.

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Algebraic Concepts: Algebraic Sets Let be the ideal generated by the finite set of polynomials F = { f 1, f 2,…, f r }. Let p = ( a 1,…, a n ) be a point in k n 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 f i ( p ) = 0 for every generator f i in F. In 3D, the algebraic surface f = 0 is the algebraic set of the ideal. source: Hoffmann

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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. source: Hoffmann* see Hoffman’s Section 7.2.6 for subtleties related to this statement.

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Algebraic Concepts: Algebraic Sets and Varieties (cont.) Given generators F = { f 1, f 2,…, f r }, the algebraic set defined by F in k n has dimension n-r –If equations f i = 0 are algebraically independent. –Complication: some of ideal’s components may have different dimensions. source: Hoffmann

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Algebraic Concepts: Algebraic Sets and Varieties (cont.) source: Hoffmann Consider algebraic set V ( I ) for ideal I in k n. 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.

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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 { f 1, g 1 }. Irreducible ellipse in plane is is algebraic set in ideal generated by { f 1, g 2 }. Ideal is reducible. –Decomposes into and Algebraic set –Varieties: V ( I 2 ) and V ( I 3 ) source: Hoffmann

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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. source: Hoffmann Bezout’s Theorem*: 2 irreducible surfaces of degree m and n intersect in a curve of degree mn. * allowing complex coordinates, points at infinity

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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 = { f 1, f 2,…, f r }, 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 h i are polynomials.” Strategy: rewrite g until original question can be easily answered. source: Hoffmann

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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: 1.x 2.If then 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

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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: source: Hoffmann

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Gröbner Bases: Rewriting and Normal-Form Algorithms Given polynomial g and set of polynomials F = { f 1, f 2,…, f r } –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

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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

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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

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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 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

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

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Gröbner Bases: Buchberger’s Theorem & Construction (continued) Buchberger’s Theorem: foundation of algorithm source: Hoffmann Gröbner basis construction algorithm

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