Elimination Theory in the 21st century

Transcription

1 NSF-CBMS Conference on Applications of Polynomial Systems

2 Before we start...

3 Before we start... Session on Open Problems Friday 3.30 pm

4 Before we start... Session on Open Problems Friday 3.30 pm BYOP

5 Before we start... Session on Open Problems Friday 3.30 pm BYOP 5 minutes per presentation

6 Before we start... Session on Open Problems Friday 3.30 pm BYOP 5 minutes per presentation to book a time slot, contact Hal or me

7 The Paradigm of the 21st Century

8 The Paradigm of the 21st Century Apart from being crazy about applications

9 The Paradigm of the 21st Century Apart from being crazy about applications and implementations...

10 The Paradigm of the 21st Century Apart from being crazy about applications and implementations...

11 Cost of Elimination

12 Cost of Elimination The number and size of the solutions of F 1 (X 1,..., X n ) = 0 F 2 (X 1,..., X n ) = 0... F n (X 1,..., X n ) = 0

13 Cost of Elimination The number and size of the solutions of F 1 (X 1,..., X n ) = 0 F 2 (X 1,..., X n ) = 0... F n (X 1,..., X n ) = 0 where size of F i = (d, L)

14 Cost of Elimination The number and size of the solutions of F 1 (X 1,..., X n ) = 0 F 2 (X 1,..., X n ) = 0... F n (X 1,..., X n ) = 0 where size of F i = (d, L) is bounded by and generically equal to ( d n, nd n 1 L )

15

16 The output is already exponential!!!

17 The output is already exponential!!! Moreover:

18 The output is already exponential!!! Moreover: the complexity of computing Gröbner bases is double exponential in (d, L)

19 What can we do then???

20 What can we do then??? Relax the input

21 What can we do then??? Relax the input (and expect the output to be relaxed too!)

22 What can we do then??? Relax the input (and expect the output to be relaxed too!) Change the Computational Model

23 Relaxing the input

24 Relaxing the input Tropical Geometry

25 Relaxing the input Tropical Geometry Relaxed resultants

26 Relaxing the input Tropical Geometry Relaxed resultants Implicitization matrices

27 Relaxing the input Tropical Geometry Relaxed resultants Implicitization matrices (Wednesday)

28 Relaxing the input Tropical Geometry Relaxed resultants Implicitization matrices (Wednesday) Syzygies and Rees Algebras

29 Relaxing the input Tropical Geometry Relaxed resultants Implicitization matrices (Wednesday) Syzygies and Rees Algebras (Wednesday)

30 Relaxing the input Tropical Geometry Relaxed resultants Implicitization matrices (Wednesday) Syzygies and Rees Algebras (Wednesday)...

31 Changing the model

32 Changing the model Probabilistic algorithms

33 Changing the model Probabilistic algorithms Computations over the Reals

34 Changing the model Probabilistic algorithms Computations over the Reals Parallelization

35 Changing the model Probabilistic algorithms Computations over the Reals Parallelization Homotopy methods

36 Changing the model Probabilistic algorithms Computations over the Reals Parallelization Homotopy methods (Tomorrow)

37 Changing the model Probabilistic algorithms Computations over the Reals Parallelization Homotopy methods (Tomorrow)...

38 Some Tapas into the 21st century

39 Some Tapas into the 21st century

40 Some Tapas into the 21st century Disclaimer

41 Some Tapas into the 21st century Disclaimer These are not the highlights of the 21st century (so far), just a sampling of results

42 Some Tapas into the 21st century Disclaimer These are not the highlights of the 21st century (so far), just a sampling of results All these areas are very active and promising right now

43 Some Tapas into the 21st century Disclaimer These are not the highlights of the 21st century (so far), just a sampling of results All these areas are very active and promising right now, more will appear in the future...

44 Some Tapas into the 21st century Disclaimer These are not the highlights of the 21st century (so far), just a sampling of results All these areas are very active and promising right now, more will appear in the future... A lot of important names and interesting areas will be omitted...

45 Some Tapas into the 21st century Disclaimer These are not the highlights of the 21st century (so far), just a sampling of results All these areas are very active and promising right now, more will appear in the future... A lot of important names and interesting areas will be omitted...sorry!

46 Tropical: from Exponential to Polynomial

47 Tropical: from Exponential to Polynomial The logarithmic limit set of a variety V C n is a fan in R n

48 Tropical: from Exponential to Polynomial The logarithmic limit set of a variety V C n is a fan in R n, the Tropical Variety of V

49 Tropical: from Exponential to Polynomial The logarithmic limit set of a variety V C n is a fan in R n, the Tropical Variety of V (MSC Txx)

50 Computing Tropical Varieties

51 Computing Tropical Varieties is fast

52 Computing Tropical Varieties is fast encodes a lot of information about the original variety

53 Computing Tropical Varieties is fast encodes a lot of information about the original variety (dimension, degree, singularities...)

54 First tapa: Tropical Discriminants

55 First tapa: Tropical Discriminants Alicia Dickenstein, Eva Maria Feitchner, Bernd Sturmfels Tropical Discriminants ( JAMS 2007)

56 Main Result

57 Main Result Theorem 1.1

58 Main Result Theorem 1.1 For any d n matrix A, the tropical A-discriminant τ(x A ) equals the Minkowski sum of the co-bergman fan B (A) and the row space of A.

59 What is the co-bergman fan?

60 What is the co-bergman fan? is the tropicalization of the kernel of A

61 Tapa # 2:Tropical Elimination

62 Tapa # 2:Tropical Elimination Bernd Sturmfels, Jenia Tevelev Elimination Theory for Tropical Varieties ( MRL 2008)

63 Theorem1.1

64 Theorem1.1 Let α : T n T d be a homomorphism of tori

65 Theorem1.1 Let α : T n T d be a homomorphism of tori and let A : Z n Z d be the corresponding linear map of lattices of one-parameter subgroups.

66 Theorem1.1 Let α : T n T d be a homomorphism of tori and let A : Z n Z d be the corresponding linear map of lattices of one-parameter subgroups. Suppose that α induces a generically finite morphism from X onto α(x ).

67 Theorem1.1 Let α : T n T d be a homomorphism of tori and let A : Z n Z d be the corresponding linear map of lattices of one-parameter subgroups. Suppose that α induces a generically finite morphism from X onto α(x ). Then A induces a generically finite map of tropical varieties from T (X ) onto T (α(x )).

68 What about Tropical Resultants?

69 What about Tropical Resultants? Anders Jensen, Josephine Yu Computing Tropical Resultants (JA 2013)

70 From the abstract:

71 From the abstract: We fix the supports A = (A 1,..., A k ) of a list of tropical polynomials and define the tropical resultant TR(A) to be the set of choices of coefficients such that the tropical polynomials have a common solution.

72 From the abstract II

73 From the abstract II We prove that TR(A) is the tropicalization of the algebraic variety of solvable systems and that its dimension can be computed in polynomial time...

74 From the abstract III

75 From the abstract III

76 From the abstract III We also present a new algorithm for recovering a Newton polytope from the support of its tropical hypersurface. We use this to compute the Newton polytope of the sparse resultant polynomial in the case when TR(A) is of codimension 1...

77 Relaxed resultants

78 Relaxed resultants In the real world systems of equations are neither homogeneous nor all the monomials appear in the expansion F 0 = a 01 + a 02 X 1 2 X a 03 X 1 X 2 3 F 1 = a 10 + a 11 X a 12 X 1 X 2 2 F 2 = a 20 X a 21 X 1 X 2

79 Sparse Resultants (after GKZ)

80 Sparse Resultants (after GKZ) D, Martín Sombra A Poisson formula for the sparse resultant (PLMS 2015) W = {(c i,a, ξ) : F i (ξ) = 0 i} P A 0... P An (C ) n π π(w ) P A 0... P An

81 Sparse Resultants (after GKZ) D, Martín Sombra A Poisson formula for the sparse resultant (PLMS 2015) W = {(c i,a, ξ) : F i (ξ) = 0 i} P A 0... P An (C ) n π π(w ) P A 0... P An The sparse resultant of F 0,..., F n is the defining equation of the direct image π W

82 Sparse Resultants (after GKZ) D, Martín Sombra A Poisson formula for the sparse resultant (PLMS 2015) W = {(c i,a, ξ) : F i (ξ) = 0 i} P A 0... P An (C ) n π π(w ) P A 0... P An The sparse resultant of F 0,..., F n is the defining equation of the direct image π W d n MV (N(A 1 ),..., N(A n ))

83 Sparse Resultants (after GKZ) D, Martín Sombra A Poisson formula for the sparse resultant (PLMS 2015) W = {(c i,a, ξ) : F i (ξ) = 0 i} P A 0... P An (C ) n π π(w ) P A 0... P An The sparse resultant of F 0,..., F n is the defining equation of the direct image π W d n MV (N(A 1 ),..., N(A n )) nd n 1 L nmv (N(A 1 ),..., N(A n ))L

84 Other relaxed resultants

85 Other relaxed resultants By replacing (C ) n with other efficient varieties X, we get efficient resultants: W = {(c i,a, ξ) : F i (ξ) = 0 i} P N 0... P Nn X π π(w ) P N 0... P Nn

86 Examples of relaxed resultants Reduced Resultants (Zariski, Jouanolou,...)

87 Examples of relaxed resultants Reduced Resultants (Zariski, Jouanolou,...) Weighted Resultants (Jouanolou)

88 Examples of relaxed resultants Reduced Resultants (Zariski, Jouanolou,...) Weighted Resultants (Jouanolou) Parametric Resultants (Busé-Elkadi-Mourrain)

89 Examples of relaxed resultants Reduced Resultants (Zariski, Jouanolou,...) Weighted Resultants (Jouanolou) Parametric Resultants (Busé-Elkadi-Mourrain) Residual Resultants (Busé)

90 Examples of relaxed resultants Reduced Resultants (Zariski, Jouanolou,...) Weighted Resultants (Jouanolou) Parametric Resultants (Busé-Elkadi-Mourrain) Residual Resultants (Busé)...

91 Implicitization matrices

92 Implicitization matrices Recall van der Waerden s matrix introduced this morning:.. M s (λ) x β = x α F i..

93 Implicitization matrices Recall van der Waerden s matrix introduced this morning:.. M s (λ) x β = x α F i.. For s 0, the rank of M s (λ) drops iff there is a common solution

94 Rank vs Determinant

95 Rank vs Determinant Laurent Busé Implicit matrix representations of rational Bézier curves and surfaces (CAGD 2014)

96 Idea Given a rational parametrization of a spatial curve (φ 1 (t), φ 2 (t), φ 3 (t))

97 Idea Given a rational parametrization of a spatial curve (φ 1 (t), φ 2 (t), φ 3 (t)) Classic implicitization:

98 Idea Given a rational parametrization of a spatial curve (φ 1 (t), φ 2 (t), φ 3 (t)) Classic implicitization: Compute the equations F i (X, Y, Z) of the image

99 Idea Given a rational parametrization of a spatial curve (φ 1 (t), φ 2 (t), φ 3 (t)) Classic implicitization: Compute the equations F i (X, Y, Z) of the image 21st century implicitization:

100 Idea Given a rational parametrization of a spatial curve (φ 1 (t), φ 2 (t), φ 3 (t)) Classic implicitization: Compute the equations F i (X, Y, Z) of the image 21st century implicitization: Compute a matrix M(X, Y, Z) such that its rank drops on the points of the curve

101 Properties

102 Properties Entries are linear in X, Y, Z

103 Properties Entries are linear in X, Y, Z Testing rank is fast by using numerical methods

104 Properties Entries are linear in X, Y, Z Testing rank is fast by using numerical methods Over the reals one can use SVD and other numerical methods

105 Properties Entries are linear in X, Y, Z Testing rank is fast by using numerical methods Over the reals one can use SVD and other numerical methods Well suited for a lot of problems in CAGD (properness, inversion,...)

106 More about this on Wednesday!

107 Syzygies and Rees Algebras

108 Syzygies and Rees Algebras Warming up tapa

109 Syzygies and Rees Algebras Warming up tapa The rest on Wednesday :-)

110 Example (Sederberg & Chen 1995)

111 Example (Sederberg & Chen 1995) The implicit equation of a rational quartic can be computed as a 2 2 determinant.

112 Example (Sederberg & Chen 1995) The implicit equation of a rational quartic can be computed as a 2 2 determinant. If the curve has a triple point, then one row is linear and the other is cubic.

113 Example (Sederberg & Chen 1995) The implicit equation of a rational quartic can be computed as a 2 2 determinant. If the curve has a triple point, then one row is linear and the other is cubic. Otherwise, both rows are quadratic.

114 A quartic with a triple point

115 A quartic with a triple point φ(t 0, t 1 ) = (t 4 0 t 4 1 : t 2 0t 2 1 : t 0 t 3 1)

116 A quartic with a triple point φ(t 0, t 1 ) = (t 4 0 t 4 1 : t 2 0t 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

117 A quartic with a triple point φ(t 0, t 1 ) = (t 4 0 t 4 1 : t 2 0t 2 1 : t 0 t 3 1) F (X 0, X 1, X 2 ) = X2 4 X1 4 X 0 X 1 X2 2 L 1,1 (T, X ) = T 0 X 2 + T 1 X 1 L 1,3 (T, X ) = T 0 (X1 3 + X 0 X2 2 ) + T 1 X2 3 ( X 2 X 1 X1 3 + X 0 X2 2 X2 3 )

118 A quartic without triple points φ(t 0 : t 1 ) = (t 4 0 : 6t 2 0t 2 1 4t 4 1 : 4t 3 0t 1 4t 0 t 3 1)

119 A quartic without triple points φ(t 0 : t 1 ) = (t 4 0 : 6t 2 0t 2 1 4t 4 1 : 4t 3 0t 1 4t 0 t 3 1) F (X ) = X X 0 X X 0 X 1 X X 2 0 X 2 1 6X 2 0 X X 3 0 X 1

120 A quartic without triple points φ(t 0 : t 1 ) = (t 4 0 : 6t 2 0t 2 1 4t 4 1 : 4t 3 0t 1 4t 0 t 3 1) F (X ) = X X 0 X X 0 X 1 X X 2 0 X 2 1 6X 2 0 X X 3 0 X 1 L 1,2 (T, X ) = T 0 (X 1 X 2 X 0 X 2 ) + T 1 ( X 2 2 2X 0 X 1 + 4X 2 0 ) L 1,2 (T, X ) = T 0 (X X 2 2 2X 0 X 1 ) + T 1 (X 0 X 2 X 1 X 2 )

121 A very compact formula

122 A very compact formula If the curve has a point of multiplicity d 1

123 A very compact formula If the curve has a point of multiplicity d 1 the implicit equation is always a 2 2 determinant of a moving line and a moving curve of degree d 1 L 1,1 (X ) L 1,1 (X ) L 1,d 1 (X ) L 1,d 1 (X )

124 The method of moving curves

125 The method of moving curves The implicit equation of a rational curve should be computed as the determinant of a small matrix whose entries are

126 The method of moving curves The implicit equation of a rational curve should be computed as the determinant of a small matrix whose entries are some moving lines some moving conics some moving cubics

127 The method of moving curves The implicit equation of a rational curve should be computed as the determinant of a small matrix whose entries are some moving lines some moving conics some moving cubics the more singular the curve, the simpler the description of the determinant

128 In general, we do not know..

129 In general, we do not know.. which moving lines? which moving conics? which moving cubics?

130 Some order in the 21st century

131 Some order in the 21st century David Cox The moving curve ideal and the Rees algebra (TCS 2008)

132 Some order in the 21st century David Cox The moving curve ideal and the Rees algebra (TCS 2008) K φ := {Moving curves following φ} = homogeneous elements in the 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 φ 0 (T )s X 1 φ 1 (T )s X 2 φ 2 (T )s

133 Some order in the 21st century David Cox The moving curve ideal and the Rees algebra (TCS 2008) K φ := {Moving curves following φ} = homogeneous elements in the 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 φ 0 (T )s X 1 φ 1 (T )s X 2 φ 2 (T )s The image of this map is the Rees Algebra of φ

134 Method of moving curves revisited

135 Method of moving curves revisited The implicit equation should be obtained as the determinant of a matrix with

136 Method of moving curves revisited The implicit equation should be obtained as the determinant of a matrix with some minimal generators of K φ and relations among them

137 Method of moving curves revisited The implicit equation should be obtained as the determinant of a matrix with some minimal generators of K φ and relations among them The more singular the curve, the simpler the description of K φ

138 21st Century Problem

139 21st Century Problem Compute a minimal system of generators of K φ

140 21st Century Problem Compute a minimal system of generators of K φ for any φ

141 21st Century Problem Compute a minimal system of generators of K φ for any φ We are working on that!

142 21st Century Problem Compute a minimal system of generators of K φ for any φ We are working on that! (More on Wednesday)

143 21st Century Problem Compute a minimal system of generators of K φ for any φ We are working on that! (More on Wednesday)

144 Changing the model of computation

145 Probabilistic Algorithms: faster they are

146 Probabilistic Algorithms: faster they are Gabriela Jeronimo, Teresa Krick, Juan Sabia, Martín Sombra The Computational Complexity of the Chow Form ( JFoCM 2004)

147 From the abstract: We present a bounded probability algorithm for the computation of the Chow forms (resultants) of the equidimensional components of a variety...

148 From the abstract: We present a bounded probability algorithm for the computation of the Chow forms (resultants) of the equidimensional components of a variety... The expected complexity of the algorithm is polynomial in the size and the geometric degree of the input equation system defining the variety...

149 From the abstract: We present a bounded probability algorithm for the computation of the Chow forms (resultants) of the equidimensional components of a variety... The expected complexity of the algorithm is polynomial in the size and the geometric degree of the input equation system defining the variety... As an application, we obtain an algorithm to compute a subclass of sparse resultants, whose complexity is polynomial in the dimension and the volume of the input set of exponents...

150 Main idea

151 Main idea Given (equations) of a d-dimensional variety V P n

152 Main idea Given (equations) of a d-dimensional variety V P n Cut V with general (probabilistically) d hyperplanes to obtain finite points: V {x 1 =... = x d = 0} = {ξ 1,..., ξ D } From (a geometric resolution of) {ξ 1,..., ξ D }, lift (Newton s Method) to the Chow Form of V {x 1 =... = x d 1 = 0}

153 Main idea Given (equations) of a d-dimensional variety V P n Cut V with general (probabilistically) d hyperplanes to obtain finite points: V {x 1 =... = x d = 0} = {ξ 1,..., ξ D } From (a geometric resolution of) {ξ 1,..., ξ D }, lift (Newton s Method) to the Chow Form of V {x 1 =... = x d 1 = 0} Do it (d 1) times more...

154 Star Model: Straight-Line Programs

155 Star Model: Straight-Line Programs Gabriela Jeronimo, Juan Sabia Sparse resultants and straight-line programs (JSC 2018)

156 What is a straight-line program?

157 What is a straight-line program? It is a program with no branches

158 What is a straight-line program? It is a program with no branches, no loops

159 What is a straight-line program? It is a program with no branches, no loops, no conditional statements

160 What is a straight-line program? It is a program with no branches, no loops, no conditional statements, no comparisons

161 What is a straight-line program? It is a program with no branches, no loops, no conditional statements, no comparisons...just a sequence of basic operations

162 From the abstract:

163 From the abstract: We prove that the sparse resultant, redefined by D Andrea and Sombra and by Esterov as a power of the classical sparse resultant, can be evaluated in a number of steps which is polynomial in its degree, its number of variables and the size of the exponents...

164 From the abstract: We prove that the sparse resultant, redefined by D Andrea and Sombra and by Esterov as a power of the classical sparse resultant, can be evaluated in a number of steps which is polynomial in its degree, its number of variables and the size of the exponents... Moreover, we design a probabilistic algorithm of this order of complexity to compute a straight-line program that evaluates it within this number of steps.

165 Main ideas

166 Main ideas SLP s

167 Main ideas SLP s Geometric Resolution

168 Main ideas SLP s Geometric Resolution Newton Hensel Lifting

169 Main ideas SLP s Geometric Resolution Newton Hensel Lifting Padé Approximations...

170 Anoter real model: BSS-machine

171 Anoter real model: BSS-machine is a Random Access Machine

172 Anoter real model: BSS-machine is a Random Access Machine with registers that can store arbitrary real numbers

173 Anoter real model: BSS-machine is a Random Access Machine with registers that can store arbitrary real numbers and that can compute rational functions over the reals at unit cost

174 Parallelize: find just one solution

175 Parallelize: find just one solution Given f (x) Z[x], compute a small

176 Parallelize: find just one solution Given f (x) Z[x], compute a small approximate root

177 Parallelize: find just one solution Given f (x) Z[x], compute a small approximate root, fast

178 What is an approximate root?

179 Steve Smale s 17th s problem (1998)

180 Steve Smale s 17th s problem (1998) Is there an algorithm which computes an approximate solution of a system of polynomials in time polynomial on the average, in the size of the input?

181 Solving Smale s 17th Problem

182 Solving Smale s 17th Problem Carlos Beltrán, Luis Miguel Pardo. On Smale s 17th Problem: A Probabilistic Positive answer ( JFoCM 2008)

183 Solving Smale s 17th Problem Carlos Beltrán, Luis Miguel Pardo. On Smale s 17th Problem: A Probabilistic Positive answer ( JFoCM 2008) Carlos Beltrán, Luis Miguel Pardo. Smale s 17th Problem: Average Polynomial Time to compute affine and projective solutions (JAMS 2009)

184 Solving Smale s 17th Problem Carlos Beltrán, Luis Miguel Pardo. On Smale s 17th Problem: A Probabilistic Positive answer ( JFoCM 2008) Carlos Beltrán, Luis Miguel Pardo. Smale s 17th Problem: Average Polynomial Time to compute affine and projective solutions (JAMS 2009) Felipe Cucker, Peter Bürgisser. On a problem posed by Steve Smale (Ann.Math 2011)

185 Solving Smale s 17th Problem Carlos Beltrán, Luis Miguel Pardo. On Smale s 17th Problem: A Probabilistic Positive answer ( JFoCM 2008) Carlos Beltrán, Luis Miguel Pardo. Smale s 17th Problem: Average Polynomial Time to compute affine and projective solutions (JAMS 2009) Felipe Cucker, Peter Bürgisser. On a problem posed by Steve Smale (Ann.Math 2011)

186 Pierre Lairez (2017)

187 Pierre Lairez (2017)

188 Pierre Lairez (2017) A deterministic algorithm to compute approximate roots of polynomial systems in polynomial average time JFoCM (2017)

189 Homotopies (In depth tomorrow!)

190 Homotopies (In depth tomorrow!) Start with an easy system

191 Homotopies (In depth tomorrow!) Start with an easy system Chase the roots with an homotopy + Newton s method

192 Numerical Algebraic Geometry

193 Numerical Algebraic Geometry By using homotopies, you can compute very fast

194 Numerical Algebraic Geometry By using homotopies, you can compute very fast Components

195 Numerical Algebraic Geometry By using homotopies, you can compute very fast Components Multiplicities

196 Numerical Algebraic Geometry By using homotopies, you can compute very fast Components Multiplicities Decomposition

197 Numerical Algebraic Geometry By using homotopies, you can compute very fast Components Multiplicities Decomposition Dimension

198 Numerical Algebraic Geometry By using homotopies, you can compute very fast Components Multiplicities Decomposition Dimension...

199 Not in the menu today...

200 Not in the menu today... Differential/Difference Elimination

201 Not in the menu today... Differential/Difference Elimination Real Elimination

202 Not in the menu today... Differential/Difference Elimination Real Elimination Applications

203 Not in the menu today... Differential/Difference Elimination Real Elimination Applications Implementations

204 Not in the menu today... Differential/Difference Elimination Real Elimination Applications Implementations...

205 Conclusion

206 Conclusion The 21st century is/will be driven by Applications

207 Conclusion The 21st century is/will be driven by Applications Fast & efficient computations

208 Conclusion The 21st century is/will be driven by Applications Fast & efficient computations Interesting challenges and problems need your help here!

209 Thanks!