An efficient probabilistic algorithm to compute the real dimension - PowerPoint PPT Presentation

An efficient probabilistic algorithm to compute the real dimension of a real algebraic set. JNCF 2014 Ivan Bannwarth 1 Mohab Safey El Din 12 1 Université Pierre et Marie Curie INRIA, POLSYS Team LIP6 - CNRS 2 Institut Universitaire de France 4th November 2014 Ivan Bannwarth Real dimension 4th November 2014 1 / 16

Real dimension Let S be the semi-algebraic set : S = { x ∈ R n | f 1 ( x ) = · · · = f p ( x ) = 0 , g 1 ( x ) > 0 , . . . , g s ( x ) > 0 } . with f 1 , . . . , f p , g 1 , . . . , g s in R [ X 1 , . . . , X n ] Definition The real dimension of S is the maximum integer d such that in generic coordinates , Interior ( π d ( S )) � = ∅ where π d : ( x 1 , . . . , x n ) �→ ( x 1 , . . . , x d ) . Ivan Bannwarth Real dimension 4th November 2014 2 / 16

Real dimension Let S be the semi-algebraic set : S = { x ∈ R n | f 1 ( x ) = · · · = f p ( x ) = 0 , g 1 ( x ) > 0 , . . . , g s ( x ) > 0 } . with f 1 , . . . , f p , g 1 , . . . , g s in R [ X 1 , . . . , X n ] Definition The real dimension of S is the maximum integer d such that in generic coordinates , Interior ( π d ( S )) � = ∅ where π d : ( x 1 , . . . , x n ) �→ ( x 1 , . . . , x d ) . S Ivan Bannwarth Real dimension 4th November 2014 2 / 16

Real dimension Let S be the semi-algebraic set : S = { x ∈ R n | f 1 ( x ) = · · · = f p ( x ) = 0 , g 1 ( x ) > 0 , . . . , g s ( x ) > 0 } . with f 1 , . . . , f p , g 1 , . . . , g s in R [ X 1 , . . . , X n ] Definition The real dimension of S is the maximum integer d such that in generic coordinates , Interior ( π d ( S )) � = ∅ where π d : ( x 1 , . . . , x n ) �→ ( x 1 , . . . , x d ) . S π 2 Ivan Bannwarth Real dimension 4th November 2014 2 / 16

Motivations Motivations Applications in computational real algebraic geometry. Ivan Bannwarth Real dimension 4th November 2014 3 / 16

Motivations Motivations Applications in computational real algebraic geometry. Computing the set of realizable sign conditions. Barone/Basu 2012 Computing a bound on the number of connected component of real algebraic sets. Barone/Basu 2013 Ivan Bannwarth Real dimension 4th November 2014 3 / 16

Motivations Motivations Applications in computational real algebraic geometry. Applications in mechanics. Ivan Bannwarth Real dimension 4th November 2014 3 / 16

Motivations Motivations Applications in computational real algebraic geometry. Applications in mechanics. Overconstraint analysis on spatial 6-link loops, Jin/Yang,2002 Ivan Bannwarth Real dimension 4th November 2014 3 / 16

State-of-the-art Let S = { x ∈ R n | f 1 ( x ) = · · · = f p ( x ) = 0 , g 1 ( x ) > 0 , . . . , g s ( x ) > 0 } with maximum degree D and real dimension d . Ivan Bannwarth Real dimension 4th November 2014 4 / 16

State-of-the-art Let S = { x ∈ R n | f 1 ( x ) = · · · = f p ( x ) = 0 , g 1 ( x ) > 0 , . . . , g s ( x ) > 0 } with maximum degree D and real dimension d . Collins’s Cylindrical Algebraic Decomposition algorithm [ ∼ 70’s] → (( s + 1 ) D ) 2 O ( n ) Ivan Bannwarth Real dimension 4th November 2014 4 / 16

State-of-the-art Let S = { x ∈ R n | f 1 ( x ) = · · · = f p ( x ) = 0 , g 1 ( x ) > 0 , . . . , g s ( x ) > 0 } with maximum degree D and real dimension d . Collins’s Cylindrical Algebraic Decomposition algorithm [ ∼ 70’s] → (( s + 1 ) D ) 2 O ( n ) Vorobjov, Basu/Pollack/Roy, Koiran’s algorithms [ ∼ 90’s] → (( s + 1 ) D ) O ( d ( n − d )) Ivan Bannwarth Real dimension 4th November 2014 4 / 16

State-of-the-art Let S = { x ∈ R n | f 1 ( x ) = · · · = f p ( x ) = 0 , g 1 ( x ) > 0 , . . . , g s ( x ) > 0 } with maximum degree D and real dimension d . Collins’s Cylindrical Algebraic Decomposition algorithm [ ∼ 70’s] → (( s + 1 ) D ) 2 O ( n ) Vorobjov, Basu/Pollack/Roy, Koiran’s algorithms [ ∼ 90’s] → (( s + 1 ) D ) O ( d ( n − d )) No information on the constant in the exponent. Ivan Bannwarth Real dimension 4th November 2014 4 / 16

State-of-the-art Let S = { x ∈ R n | f 1 ( x ) = · · · = f p ( x ) = 0 , g 1 ( x ) > 0 , . . . , g s ( x ) > 0 } with maximum degree D and real dimension d . Collins’s Cylindrical Algebraic Decomposition algorithm [ ∼ 70’s] → (( s + 1 ) D ) 2 O ( n ) Vorobjov, Basu/Pollack/Roy, Koiran’s algorithms [ ∼ 90’s] → (( s + 1 ) D ) O ( d ( n − d )) No information on the constant in the exponent. There is no efficient implementation today. Ivan Bannwarth Real dimension 4th November 2014 4 / 16

State-of-the-art Let S = { x ∈ R n | f 1 ( x ) = · · · = f p ( x ) = 0 , g 1 ( x ) > 0 , . . . , g s ( x ) > 0 } with maximum degree D and real dimension d . Collins’s Cylindrical Algebraic Decomposition algorithm [ ∼ 70’s] → (( s + 1 ) D ) 2 O ( n ) Best implementation but limited ( n ≤ 3 for non-trivial examples). Vorobjov, Basu/Pollack/Roy, Koiran’s algorithms [ ∼ 90’s] → (( s + 1 ) D ) O ( d ( n − d )) No information on the constant in the exponent. There is no efficient implementation today. Ivan Bannwarth Real dimension 4th November 2014 4 / 16

Contribution Contribution New algorithm for hypersurfaces V R ( f ) (defined by f = 0) 1 General Observation In the real case, ⇒ f 2 1 ( x ) + · · · + f 2 f 1 ( x ) = · · · = f p ( x ) = 0 ⇐ p ( x ) = 0 Ivan Bannwarth Real dimension 4th November 2014 5 / 16

Contribution Contribution New algorithm for hypersurfaces V R ( f ) (defined by f = 0) 1 Best known complexity class : 2 � D 3d ( n − d )+ 6 n + 3 � � O Input : f : a polynomial of degree D d : the real dimension of V R ( f ) . Ivan Bannwarth Real dimension 4th November 2014 5 / 16

Contribution Contribution New algorithm for hypersurfaces V R ( f ) (defined by f = 0) 1 Best known complexity class : 2 � D 3d ( n − d )+ 6 n + 3 � � O Probabilistic algorithm 3 Probabilistic subroutines → Generic change of variables → One point per connected components and test of emptiness. Ivan Bannwarth Real dimension 4th November 2014 5 / 16

Contribution Contribution New algorithm for hypersurfaces V R ( f ) (defined by f = 0) 1 Best known complexity class : 2 � D 3d ( n − d )+ 6 n + 3 � � O Probabilistic algorithm 3 Efficient implementation 4 Checking procedures Grobner basis instead of geometric resolution Example reached : n = 6, D = 8, 130 sec. Ivan Bannwarth Real dimension 4th November 2014 5 / 16

Previous approach : Quantifier Elimination (QE) ∃ Z ∈ R , X 2 + Y 2 + Z 2 − 1 = 0 S π 2 Compute Φ quantifier free formula defining π i ( S ) . 1 X 2 + Y 2 − 1 ≤ 0 π 2 ( S ) Ivan Bannwarth Real dimension 4th November 2014 6 / 16

Previous approach : Quantifier Elimination (QE) ∃ Z ∈ R , X 2 + Y 2 + Z 2 − 1 = 0 S π 2 Compute Φ quantifier free formula defining π i ( S ) . 1 X 2 + Y 2 − 1 ≤ 0 π 2 ( S ) Compute ˜ Φ with strict inequalities defining an open 2 dense semi-algebraic subset of π i ( S ) . Ivan Bannwarth Real dimension 4th November 2014 6 / 16

Previous approach : Quantifier Elimination (QE) ∃ Z ∈ R , X 2 + Y 2 + Z 2 − 1 = 0 S π 2 Compute Φ quantifier free formula defining π i ( S ) . 1 X 2 + Y 2 − 1 ≤ 0 π 2 ( S ) Compute ˜ Φ with strict inequalities defining an open 2 dense semi-algebraic subset of π i ( S ) . Test if this set is empty. 3 Ivan Bannwarth Real dimension 4th November 2014 6 / 16

Previous approach : Quantifier Elimination (QE) ∃ Z ∈ R , X 2 + Y 2 + Z 2 − 1 = 0 S π 2 Compute Φ quantifier free formula defining π i ( S ) . 1 X 2 + Y 2 − 1 ≤ 0 Compute ˜ π 2 ( S ) Φ with strict inequalities defining an open 2 dense semi-algebraic subset of π i ( S ) . Test if this set is empty. 3 New appoach : Variant of QE (Hong/Safey 12) Compute Boundary ( π i ( S )) . 1 S π 2 → Hypotheses : the algebraic variety assoc. to the polynomial equations is smooth and equidimensional, the projection of S is proper. Ivan Bannwarth Real dimension 4th November 2014 6 / 16

Previous approach : Quantifier Elimination (QE) ∃ Z ∈ R , X 2 + Y 2 + Z 2 − 1 = 0 S π 2 Compute Φ quantifier free formula defining π i ( S ) . 1 X 2 + Y 2 − 1 ≤ 0 Compute ˜ π 2 ( S ) Φ with strict inequalities defining an open 2 dense semi-algebraic subset of π i ( S ) . Test if this set is empty. 3 New appoach : Variant of QE (Hong/Safey 12) Compute Boundary ( π i ( S )) . 1 S π 2 Compute one point per connected component. 2 → Hypotheses : the algebraic variety assoc. to the polynomial equations is smooth and equidimensional, the projection of S is proper. Ivan Bannwarth Real dimension 4th November 2014 6 / 16

Previous approach : Quantifier Elimination (QE) ∃ Z ∈ R , X 2 + Y 2 + Z 2 − 1 = 0 S π 2 Compute Φ quantifier free formula defining π i ( S ) . 1 X 2 + Y 2 − 1 ≤ 0 Compute ˜ π 2 ( S ) Φ with strict inequalities defining an open 2 dense semi-algebraic subset of π i ( S ) . Test if this set is empty. 3 New appoach : Variant of QE (Hong/Safey 12) Compute Boundary ( π i ( S )) . 1 S π 2 Compute one point per connected component. 2 Lift the fibers. 3 → Hypotheses : the algebraic variety assoc. to the polynomial equations is smooth and equidimensional, the projection of S is proper. Ivan Bannwarth Real dimension 4th November 2014 6 / 16