Outlines Elimination Theory, Commutative Algebra and Applications Laurent Bus´e INRIA Sophia-Antipolis, France March 2, 2005 Laurent Bus´e Elimination Theory, Commutative Algebra and Applications Part I: Residual resultants in 2 Outlines P Part II: Implicitization Problem Part I: Solving polynomial systems with resultants 1 Overview of the classical approach The Macaulay resultant Solving zero-dimensional polynomial system Limitations 2 The residual resultant approach The residual resultant Solving polynomial systems An example: cylinders passing through 5 points Laurent Bus´e Elimination Theory, Commutative Algebra and Applications Part I: Residual resultants in 2 Outlines P Part II: Implicitization Problem Part II: Implicitization problem for curves and surfaces 3 Implicitization of rational plane curves Using resultants Using moving lines Using syzygies 4 Implicitization of rational surfaces Using resultants Using syzygies Laurent Bus´e Elimination Theory, Commutative Algebra and Applications Classical approach Residual approach Part I Solving polynomial systems with resultants Laurent Bus´e Elimination Theory, Commutative Algebra and Applications The Macaulay resultant Classical approach Solving zero-dimensional polynomial system Residual approach Limitations The Macaulay resultant Consider a system of 3 homogeneous polynomials in X0, X1, X2: P α f0 = c0,α.X |α|=d0 f = P c .X α 1 |α|=d1 1,α f = P c .X α 2 |α|=d2 2,α α α0 α1 α2 X denotes a monomial X0 X1 X2 (αi ≥ 0). The ci,α’s are the parameters of the system with value in an algebraically closed field k. Theorem (Macaulay - 1902) There exists an irreducible and homogeneous polynomial in k[ci,α], denoted Res(f0, f1, f2), satisfying: 2 Res(f0, f1, f2) = 0 ⇔ ∃x ∈ Pk such that f0(x) = f1(x) = f2(x) = 0. Laurent Bus´e Elimination Theory, Commutative Algebra and Applications The Macaulay resultant Classical approach Solving zero-dimensional polynomial system Residual approach Limitations The Macaulay resultant Consider a system of 3 homogeneous polynomials in X0, X1, X2: P α f0 = c0,α.X |α|=d0 f = P c .X α 1 |α|=d1 1,α f = P c .X α 2 |α|=d2 2,α α α0 α1 α2 X denotes a monomial X0 X1 X2 (αi ≥ 0). The ci,α’s are the parameters of the system with value in an algebraically closed field k. Theorem (Macaulay - 1902) There exists an irreducible and homogeneous polynomial in k[ci,α], denoted Res(f0, f1, f2), satisfying: 2 Res(f0, f1, f2) = 0 ⇔ ∃x ∈ Pk such that f0(x) = f1(x) = f2(x) = 0. Laurent Bus´e Elimination Theory, Commutative Algebra and Applications The Macaulay resultant Classical approach Solving zero-dimensional polynomial system Residual approach Limitations Computing the Macaulay resultant Degrees: Res(f0, f1, f2) is homogeneous w.r.t. each fi : d d d deg = 0 1 2 with i ∈ {0, 1, 2} fixed. ci,α di Computation: Let ν := d0 + d1 + d2 − 2, set A := k[ci,α] and consider the map (the first map of a Koszul complex) 3 M A[X0, X1, X2]ν−di → A[X0, X1, X2]ν i=1 (g0, g1, g2) 7→ g0f0 + g1f1 + g2f2. It depends on the ci,α’s and has maximal rank Some maximal minors are the Macaulay matrices. Their gcd (3 are sufficients) gives Res(f0, f1, f2). Laurent Bus´e Elimination Theory, Commutative Algebra and Applications The Macaulay resultant Classical approach Solving zero-dimensional polynomial system Residual approach Limitations Computing the Macaulay resultant Degrees: Res(f0, f1, f2) is homogeneous w.r.t. each fi : d d d deg = 0 1 2 with i ∈ {0, 1, 2} fixed. ci,α di Computation: Let ν := d0 + d1 + d2 − 2, set A := k[ci,α] and consider the map (the first map of a Koszul complex) 3 M A[X0, X1, X2]ν−di → A[X0, X1, X2]ν i=1 (g0, g1, g2) 7→ g0f0 + g1f1 + g2f2. It depends on the ci,α’s and has maximal rank Some maximal minors are the Macaulay matrices. Their gcd (3 are sufficients) gives Res(f0, f1, f2). Laurent Bus´e Elimination Theory, Commutative Algebra and Applications The Macaulay resultant Classical approach Solving zero-dimensional polynomial system Residual approach Limitations Computing the Macaulay resultant Degrees: Res(f0, f1, f2) is homogeneous w.r.t. each fi : d d d deg = 0 1 2 with i ∈ {0, 1, 2} fixed. ci,α di Computation: Let ν := d0 + d1 + d2 − 2, set A := k[ci,α] and consider the map (the first map of a Koszul complex) 3 M A[X0, X1, X2]ν−di → A[X0, X1, X2]ν i=1 (g0, g1, g2) 7→ g0f0 + g1f1 + g2f2. It depends on the ci,α’s and has maximal rank Some maximal minors are the Macaulay matrices. Their gcd (3 are sufficients) gives Res(f0, f1, f2). Laurent Bus´e Elimination Theory, Commutative Algebra and Applications The Macaulay resultant Classical approach Solving zero-dimensional polynomial system Residual approach Limitations Recovering the Chow form V (f1, f2) Let f1, f2 ∈ R = k[X0, X1, X2] be homo- geneous polynomials defining isolated 2 points (a complete intersection) in P . Introduce L(X) = c0X0 + c1X1 + c2X2. L(X ) Proposition (Chow form) Y µ(ξ) Res(L, f1, f2) = L(ξ) , ξ∈V (f1,f2) where µ(ξ) denotes the multiplicity (or degree) of ξ. ⇒ An Absolute factorisation problem Laurent Bus´e Elimination Theory, Commutative Algebra and Applications The Macaulay resultant Classical approach Solving zero-dimensional polynomial system Residual approach Limitations Recovering the Chow form V (f1, f2) Let f1, f2 ∈ R = k[X0, X1, X2] be homo- geneous polynomials defining isolated 2 points (a complete intersection) in P . Introduce L(X) = c0X0 + c1X1 + c2X2. L(X ) Proposition (Chow form) Y µ(ξ) Res(L, f1, f2) = L(ξ) , ξ∈V (f1,f2) where µ(ξ) denotes the multiplicity (or degree) of ξ. ⇒ An Absolute factorisation problem Laurent Bus´e Elimination Theory, Commutative Algebra and Applications The Macaulay resultant Classical approach Solving zero-dimensional polynomial system Residual approach Limitations Recovering multiplication maps And one may assume that there is The Macaulay matrix with no solutions on X = 0: right degree in L : 0 EF EF . A B . X0E L f1, f2 Rν − − − . Rν \ X0E . C D Proposition The matrix of the multiplication by L in R/(f1, f2)|X0=1 is given by: −1 ML = (A − BD C)|X0=1. ⇒ The roots ξ are obtained through eigen-computations. Laurent Bus´e Elimination Theory, Commutative Algebra and Applications The Macaulay resultant Classical approach Solving zero-dimensional polynomial system Residual approach Limitations Recovering multiplication maps And one may assume that there is The Macaulay matrix with no solutions on X = 0: right degree in L : 0 EF EF . A B . X0E L f1, f2 Rν − − − . Rν \ X0E . C D Proposition The matrix of the multiplication by L in R/(f1, f2)|X0=1 is given by: −1 ML = (A − BD C)|X0=1. ⇒ The roots ξ are obtained through eigen-computations. Laurent Bus´e Elimination Theory, Commutative Algebra and Applications The Macaulay resultant Classical approach Solving zero-dimensional polynomial system Residual approach Limitations Recovering multiplication maps And one may assume that there is The Macaulay matrix with no solutions on X = 0: right degree in L : 0 EF EF . A B . X0E L f1, f2 Rν − − − . Rν \ X0E . C D Proposition The matrix of the multiplication by L in R/(f1, f2)|X0=1 is given by: −1 ML = (A − BD C)|X0=1. ⇒ The roots ξ are obtained through eigen-computations. Laurent Bus´e Elimination Theory, Commutative Algebra and Applications The Macaulay resultant Classical approach Solving zero-dimensional polynomial system Residual approach Limitations The limitations Problems: If V (f1, f2) is not finite then Res(L, f1, f2) is identically zero. If V (f1, f2) has base points (roots which are independant of the parameters ci,α’s) then Res(L, f1, f2) is also identically zero. Questions: How to remove the non finite part of V (f1, f2)? More generally, how to compute “a part” (even zero-dimensional) of V (f1, f2)? Laurent Bus´e Elimination Theory, Commutative Algebra and Applications The Macaulay resultant Classical approach Solving zero-dimensional polynomial system Residual approach Limitations The limitations Problems: If V (f1, f2) is not finite then Res(L, f1, f2) is identically zero. If V (f1, f2) has base points (roots which are independant of the parameters ci,α’s) then Res(L, f1, f2) is also identically zero. Questions: How to remove the non finite part of V (f1, f2)? More generally, how to compute “a part” (even zero-dimensional) of V (f1, f2)? Laurent Bus´e Elimination Theory, Commutative Algebra and Applications The Macaulay resultant Classical approach Solving zero-dimensional polynomial system Residual approach Limitations Three circles in the projective plane Consider the following system: 2 2 2 f0 = c0,1X0 + c0,2X0X1 + c0,3X0X2 + c0,4(X1 + X2 ) 2 2 2 f1 = c1,1X0 + c1,2X0X1 + c1,3X0X2 + c1,4(X1 + X2 ) 2 2 2 f2 = c2,1X0 + c2,2X0X1 + c2,3X0X2 + c2,4(X1 + X2 ) Problem: When do they have a common point ? f1 Two “base points”: (0 : 1 : ±i). f0 This implies Res(f0, f1, f2) ≡ 0. Refined question: When do they have a common root which is not f2 (0 : 1 : ±i)? 2 P Laurent Bus´e Elimination Theory, Commutative Algebra and Applications The Macaulay resultant Classical approach Solving zero-dimensional polynomial system Residual approach Limitations Three circles in the projective plane Consider the following system: 2 2 2 f0 = c0,1X0 + c0,2X0X1 + c0,3X0X2 + c0,4(X1 + X2 ) 2 2 2 f1 = c1,1X0 + c1,2X0X1 + c1,3X0X2 + c1,4(X1 + X2 ) 2 2 2 f2 = c2,1X0 + c2,2X0X1 + c2,3X0X2 + c2,4(X1 + X2 ) Problem: When do they have a common point ? f1 Two “base points”: (0 : 1 : ±i).