EACA 2014 XIV Encuentro de lgebra Computacional y Aplicaciones - PowerPoint PPT Presentation

EACA 2014 XIV Encuentro de Álgebra Computacional y Aplicaciones Barcelona June 18–20 2014 http://www.ub.edu/eaca2014/ Carlos D’Andrea Moving surfaces ideals of rational parametrizations

Moving surfaces ideals of rational parametrizations Carlos D’Andrea Computer Algebra and Polynomials Linz – November 2013 Carlos D’Andrea Moving surfaces ideals of rational parametrizations

Episode 1 : Curves Carlos D’Andrea Moving surfaces ideals of rational parametrizations

Rational Plane Curves P 1 P 2 φ : → ( t 0 : t 1 ) �→ ( a ( t 0 , t 1 ) : b ( t 0 , t 1 ) : c ( t 0 , t 1 )) a , b , c ∈ K [ T 0 , T 1 ] , homogeneous of the same degree d ≥ 1 gcd ( a , b , c ) = 1 Carlos D’Andrea Moving surfaces ideals of rational parametrizations

Rational Plane Curves The image of the map is a rational plane curve It has degree d if φ is injective “almost everywhere” It has genus 0, which means “maximal” number of multiple points ( d − 1 )( d − 2 ) 2 Computing the “implicit equation” is relatively easy from φ Carlos D’Andrea Moving surfaces ideals of rational parametrizations

From affine to projective K 2 K ��� � � 1 − t 2 2 t t �− → 1 + t 2 , 1 + t 2 P 1 P 2 φ : − → � t 2 0 + t 2 1 : t 2 0 − t 2 � ( t 0 : t 1 ) �− → 1 : 2 t 0 t 1 Carlos D’Andrea Moving surfaces ideals of rational parametrizations

From parametric to implicit � � Res T X 2 · a ( T ) − X 0 · c ( T ) , X 2 · b ( T ) − X 1 · c ( T ) = − 4 X 2 2 ( X 2 0 − X 2 1 − X 2 2 ) Carlos D’Andrea Moving surfaces ideals of rational parametrizations

The Sylvester Resultant X 2 T 2 0 − 2 X 0 T 0 T 1 + X 2 T 2 X 2 a ( T ) − X 0 c ( T ) = 1 X 2 T 2 0 − 2 X 1 T 0 T 1 − X 2 T 2 X 2 b ( T ) − X 1 c ( T ) = 1 � � Res T X 2 · a ( T ) − X 0 · c ( T ) , X 2 · b ( T ) − X 1 · c ( T ) =   X 2 − 2 X 0 X 2 0 0 − 2 X 0 X 2 X 2   det   − 2 X 1 − X 2 0 X 2   0 − 2 X 1 − X 2 X 2 Carlos D’Andrea Moving surfaces ideals of rational parametrizations

The Sylvester matrix is a matrix of moving lines L ( T 0 , T 1 , X 0 , X 1 , X 2 ) = v 0 ( T ) X 0 + v 1 ( T ) X 1 + v 2 ( T ) X 2 such that L ( T 0 , T 1 , u 0 ( T ) , u 1 ( T ) , u 2 ( T )) = 0 Carlos D’Andrea Moving surfaces ideals of rational parametrizations

In our example... L 1 ( T , X ) = − 2 T 2 0 T 1 X 0 + 0 X 1 + ( T 3 0 + T 0 T 2 1 ) X 2 L 2 ( T , X ) = − 2 T 0 T 2 1 X 0 + 0 X 1 + ( T 2 0 T 1 + T 3 1 ) X 2 0 X 0 − 2 T 2 0 T 1 X 1 + ( T 3 0 − T 0 T 2 L 3 ( T , X ) = 1 ) X 2 0 X 0 − 2 T 0 T 2 1 X 1 + ( T 2 0 T 1 − T 3 L 4 ( T , X ) = 1 ) X 2   X 2 − 2 X 0 X 2 0 0 X 2 − 2 X 0 X 2     X 2 − 2 X 1 − X 2 0   0 X 2 − 2 X 1 − X 2 Carlos D’Andrea Moving surfaces ideals of rational parametrizations

Can you do it in degree 1 ? L 1 ( T , X ) = T 0 − ( X 0 + X 1 ) T 1 X 2 L 2 ( T , X ) = ( − X 0 + X 1 ) T 0 + X 2 T 1 � X 2 � − X 0 − X 1 = X 2 1 + X 2 2 − X 2 det 0 − X 0 + X 1 X 2 Carlos D’Andrea Moving surfaces ideals of rational parametrizations

The module of moving lines following φ (Hilbert) There are µ ≤ d 2 and P µ ( T , X ) , Q d − µ ( T , X ) independent moving lines following φ such that every other moving line L δ ( T , X ) following φ is of the form p δ − µ ( T ) P µ ( T , X ) + q δ − d + µ ( T ) P d − µ ( T , X ) Carlos D’Andrea Moving surfaces ideals of rational parametrizations

Geometric version There exist µ ≤ d 2 and two other plane parametrizations ϕ µ ( t 0 , t 1 ) , ψ d − µ ( t 0 , t 1 ) of degrees µ, d − µ such that φ ( t 0 , t 1 ) = ϕ µ ( t 0 , t 1 ) ∧ ψ d − µ ( t 0 , t 1 ) Carlos D’Andrea Moving surfaces ideals of rational parametrizations

For the unit circle ϕ 1 ( t 0 : t 1 ) = ( − t 1 : − t 1 : t 0 ) ψ 1 ( t 0 : t 1 ) = ( − t 0 : t 0 : t 1 ) � � e 0 e 1 e 2 � � � � � − t 2 0 − t 2 1 , t 2 1 − t 2 � − t 1 − t 1 t 0 = 0 , − 2 t 0 t 1 � � � � − t 0 t 0 t 1 � � Carlos D’Andrea Moving surfaces ideals of rational parametrizations

µ bases and Hilbert’s Syzygy Theorem The homogeneous polynomial ideal � � I = a ( T ) , b ( T ) , c ( T ) ⊂ K [ T 0 , T 1 ] has a Hilbert-Burch resolution of the type 0 → K [ T ] 2 ( ϕ µ ,ψ d − µ ) t K [ T ] 3 ( a , b , c ) − → − → K [ T ] A µ basis of the parametrization is a basis of Syz ( I ) as a K [ T ] -module Carlos D’Andrea Moving surfaces ideals of rational parametrizations

Why do we care about µ bases? Implicit equation = � � Res T P µ ( T , X ) , Q d − µ ( T , X ) Busé-D (2012) If B denotes a Bézout matrix and S a Sylvester matrix then, X 2 S ( P µ ( T , X ) , Q d − µ ( T , X )) = M B ( aX 2 − cX 0 , bX 2 − cX 1 ) , with M ∈ K d × d invertible Carlos D’Andrea Moving surfaces ideals of rational parametrizations

A bit of history on the CAGD side Sederberg, Saito, Qi, Klimaszewski. (1994), Curve implicitization using moving lines , Computer Aided Geometric Design 11, 687–706 Sederberg, Chen. Implicitization using moving curves and surfaces. Proceedings of SIGGRAPH 1995, 301–308. Sederberg, Goldman, Du. (1997), Implicitizing rational curves by the method of moving algebraic curves, J. Symbolic Comp. 23, 153–175 Cox., Sederberg, Chen. (1998), The moving line ideal basis for planar rational curves , Computer Aided Geometric Design 15, 803–827 · · · Carlos D’Andrea Moving surfaces ideals of rational parametrizations

Moving conics, Moving cubics,... a j ( T ) X 2 0 + b j ( T ) X 0 X 1 + c j ( T ) X 0 X 2 + d j ( T ) X 2 1 + e j ( T ) X 1 X 2 + f j ( T ) X 2 2 is a moving conic which follows the parametrization if a j ( T ) a ( T ) 2 + b j ( T ) a ( T ) b ( T ) + c j ( T ) a ( T ) c ( T ) + d j ( T ) b ( T ) 2 + e j ( T ) b ( T ) c ( T ) + f j ( T ) c ( T ) 2 = 0 Carlos D’Andrea Moving surfaces ideals of rational parametrizations

The method of moving curves for implicitization The implicit equation may be computed as a small determinant of � � some moving lines � � � � some moving conics � � � � some moving cubics � � � � · · · � � the more singular the curve is, the smaller the determinant is Carlos D’Andrea Moving surfaces ideals of rational parametrizations

One of those theorems (Sederberg & Chen 1995) The implicit equation of a quartic curve with no base points can be written as a 2 × 2 determinant. If the curve doesn’t have a triple point, then each element of the determinant is a quadratic; otherwise one row is linear and one row is cubic Carlos D’Andrea Moving surfaces ideals of rational parametrizations

A quartic with triple point φ ( t 0 , t 1 ) = ( t 4 0 − t 4 1 : − t 2 0 t 2 1 : t 0 t 3 1 ) F ( X 0 , X 1 , X 2 ) = X 4 2 − X 4 1 − X 0 X 1 X 2 2 L 1 , 1 ( T , X ) = T 0 X 2 + T 1 X 1 L 1 , 3 ( T , X ) = T 0 ( X 3 1 + X 0 X 2 2 ) + T 1 X 3 2 � � X 2 X 1 X 3 1 + X 0 X 2 X 3 2 2 Carlos D’Andrea Moving surfaces ideals of rational parametrizations

A quartic without a triple point φ ( t 0 : t 1 ) = ( t 4 0 : 6 t 2 0 t 2 1 − 4 t 4 1 : 4 t 3 0 t 1 − 4 t 0 t 3 1 ) F ( X ) = X 4 2 + 4 X 0 X 3 1 + 2 X 0 X 1 X 2 2 − 16 X 2 0 X 2 1 − 6 X 2 0 X 2 2 + 16 X 3 0 X 1 T 0 ( X 1 X 2 − X 0 X 2 ) + T 1 ( − X 2 2 − 2 X 0 X 1 + 4 X 2 L 1 , 2 ( T , X ) = 0 ) ˜ T 0 ( X 2 1 + 1 2 X 2 L 1 , 2 ( T , X ) = 2 − 2 X 0 X 1 ) + T 1 ( X 0 X 2 − X 1 X 2 ) Carlos D’Andrea Moving surfaces ideals of rational parametrizations

For large d , we do not know... � � which moving lines? � � � � which moving conics? � � � � which moving cubics? � � � � · · · � � Carlos D’Andrea Moving surfaces ideals of rational parametrizations

The Rees Algebra associated to the parametrization Cox, D. The moving curve ideal and the Rees algebra. Theoret. Comput. Sci. 392 (2008), no. 1–3, 23–36. K φ := { moving curves following φ } = kernel of K [ T 0 , T 1 , X 0 , X 1 , X 2 ] → K [ T 0 , T 1 , s ] T i �→ T i X 0 �→ a ( T ) s X 1 �→ b ( T ) s X 2 �→ c ( T ) s “The defining ideal of the Rees Algebra associated to φ ” Carlos D’Andrea Moving surfaces ideals of rational parametrizations

The implicit equation may be obtained as the determinant of a very small matrix: � � · · · � � � � some minimal generators of K φ � � � � · · · � � The more singular the curve, the simpler the description of K φ Carlos D’Andrea Moving surfaces ideals of rational parametrizations

Main Problem Compute a minimal set of generators of K φ for any φ Known for: µ = 1 (Hong-Simis-Vasconcelos, Cox-Hoffmann-Wang, Busé, Cortadellas- D ) µ = 2(Busé, Cortadellas- D , Kustin-Polini-Ulrich) � � K φ ( 1 , 2 ) � = 0 (Cortadellas- D ) Monomial plane parametrizations (Cortadellas- D ) Carlos D’Andrea Moving surfaces ideals of rational parametrizations

Coarse problem Compute n 0 ( K φ ) , the number of minimal generators of K φ Show that if φ is “more singular” than φ ′ then n 0 ( K φ ) ≤ n 0 ( K φ ′ ) Carlos D’Andrea Moving surfaces ideals of rational parametrizations