Elimination Theory in the 21st century
Carlos D’Andrea
NSF-CBMS Conference on Applications of Polynomial Systems
Carlos D’Andrea Elimination Theory in the 21st century Session on Open Problems Friday 3.30 pm BYOP 5 minutes per presentation to book a time slot, contact Hal or me
Before we start...
Carlos D’Andrea Elimination Theory in the 21st century BYOP 5 minutes per presentation to book a time slot, contact Hal or me
Before we start...
Session on Open Problems Friday 3.30 pm
Carlos D’Andrea Elimination Theory in the 21st century 5 minutes per presentation to book a time slot, contact Hal or me
Before we start...
Session on Open Problems Friday 3.30 pm BYOP
Carlos D’Andrea Elimination Theory in the 21st century to book a time slot, contact Hal or me
Before we start...
Session on Open Problems Friday 3.30 pm BYOP 5 minutes per presentation
Carlos D’Andrea Elimination Theory in the 21st century 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
Carlos D’Andrea Elimination Theory in the 21st century Apart from being crazy about applications and implementations...
The Paradigm of the 21st Century
Carlos D’Andrea Elimination Theory in the 21st century and implementations...
The Paradigm of the 21st Century Apart from being crazy about applications
Carlos D’Andrea Elimination Theory in the 21st century The Paradigm of the 21st Century Apart from being crazy about applications and implementations...
Carlos D’Andrea Elimination Theory in the 21st century The Paradigm of the 21st Century Apart from being crazy about applications and implementations...
Carlos D’Andrea Elimination Theory in the 21st century The number and “size” of the solutions of F1(X1,..., Xn) = 0 F2(X1,..., Xn) = 0 . . . . . . Fn(X1,..., Xn) = 0 where “size” of Fi = (d, L) is bounded by and generically equal to
d n, nd n−1L
“Cost” of Elimination
Carlos D’Andrea Elimination Theory in the 21st century where “size” of Fi = (d, L) is bounded by and generically equal to
d n, nd n−1L
“Cost” of Elimination
The number and “size” of the solutions of F1(X1,..., Xn) = 0 F2(X1,..., Xn) = 0 . . . . . . Fn(X1,..., Xn) = 0
Carlos D’Andrea Elimination Theory in the 21st century is bounded by and generically equal to
d n, nd n−1L
“Cost” of Elimination
The number and “size” of the solutions of F1(X1,..., Xn) = 0 F2(X1,..., Xn) = 0 . . . . . . Fn(X1,..., Xn) = 0 where “size” of Fi = (d, L)
Carlos D’Andrea Elimination Theory in the 21st century “Cost” of Elimination
The number and “size” of the solutions of F1(X1,..., Xn) = 0 F2(X1,..., Xn) = 0 . . . . . . Fn(X1,..., Xn) = 0 where “size” of Fi = (d, L) is bounded by and generically equal to
d n, nd n−1L
Carlos D’Andrea Elimination Theory in the 21st century Carlos D’Andrea Elimination Theory in the 21st century Moreover: the complexity of computing Gr¨obnerbases is double exponential in( d, L)
The output is already exponential!!!
Carlos D’Andrea Elimination Theory in the 21st century the complexity of computing Gr¨obnerbases is double exponential in( d, L)
The output is already exponential!!!
Moreover:
Carlos D’Andrea Elimination Theory in the 21st century The output is already exponential!!!
Moreover: the complexity of computing Gr¨obnerbases is double exponential in( d, L)
Carlos D’Andrea Elimination Theory in the 21st century “Relax” the input (and expect the output to be “relaxed” too!) Change the Computational Model
What can we do then???
Carlos D’Andrea Elimination Theory in the 21st century (and expect the output to be “relaxed” too!) Change the Computational Model
What can we do then???
“Relax” the input
Carlos D’Andrea Elimination Theory in the 21st century Change the Computational Model
What can we do then???
“Relax” the input (and expect the output to be “relaxed” too!)
Carlos D’Andrea Elimination Theory in the 21st century What can we do then???
“Relax” the input (and expect the output to be “relaxed” too!) Change the Computational Model
Carlos D’Andrea Elimination Theory in the 21st century Tropical Geometry “Relaxed” resultants Implicitization matrices (Wednesday) Syzygies and Rees Algebras (Wednesday) ...
Relaxing the input
Carlos D’Andrea Elimination Theory in the 21st century “Relaxed” resultants Implicitization matrices (Wednesday) Syzygies and Rees Algebras (Wednesday) ...
Relaxing the input
Tropical Geometry
Carlos D’Andrea Elimination Theory in the 21st century Implicitization matrices (Wednesday) Syzygies and Rees Algebras (Wednesday) ...
Relaxing the input
Tropical Geometry “Relaxed” resultants
Carlos D’Andrea Elimination Theory in the 21st century (Wednesday) Syzygies and Rees Algebras (Wednesday) ...
Relaxing the input
Tropical Geometry “Relaxed” resultants Implicitization matrices
Carlos D’Andrea Elimination Theory in the 21st century Syzygies and Rees Algebras (Wednesday) ...
Relaxing the input
Tropical Geometry “Relaxed” resultants Implicitization matrices (Wednesday)
Carlos D’Andrea Elimination Theory in the 21st century (Wednesday) ...
Relaxing the input
Tropical Geometry “Relaxed” resultants Implicitization matrices (Wednesday) Syzygies and Rees Algebras
Carlos D’Andrea Elimination Theory in the 21st century ...
Relaxing the input
Tropical Geometry “Relaxed” resultants Implicitization matrices (Wednesday) Syzygies and Rees Algebras (Wednesday) Carlos D’Andrea Elimination Theory in the 21st century Relaxing the input
Tropical Geometry “Relaxed” resultants Implicitization matrices (Wednesday) Syzygies and Rees Algebras (Wednesday) ... Carlos D’Andrea Elimination Theory in the 21st century Probabilistic algorithms Computations “over the Reals” “Parallelization” Homotopy methods (Tomorrow) ...
Changing the model
Carlos D’Andrea Elimination Theory in the 21st century Computations “over the Reals” “Parallelization” Homotopy methods (Tomorrow) ...
Changing the model
Probabilistic algorithms
Carlos D’Andrea Elimination Theory in the 21st century “Parallelization” Homotopy methods (Tomorrow) ...
Changing the model
Probabilistic algorithms Computations “over the Reals”
Carlos D’Andrea Elimination Theory in the 21st century Homotopy methods (Tomorrow) ...
Changing the model
Probabilistic algorithms Computations “over the Reals” “Parallelization”
Carlos D’Andrea Elimination Theory in the 21st century (Tomorrow) ...
Changing the model
Probabilistic algorithms Computations “over the Reals” “Parallelization” Homotopy methods
Carlos D’Andrea Elimination Theory in the 21st century ...
Changing the model
Probabilistic algorithms Computations “over the Reals” “Parallelization” Homotopy methods (Tomorrow)
Carlos D’Andrea Elimination Theory in the 21st century Changing the model
Probabilistic algorithms Computations “over the Reals” “Parallelization” Homotopy methods (Tomorrow) ... Carlos D’Andrea Elimination Theory in 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!
Some “Tapas” into the 21st century
Carlos D’Andrea Elimination Theory in 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!
Some “Tapas” into the 21st century
Carlos D’Andrea Elimination Theory in the 21st century 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!
Some “Tapas” into the 21st century
Disclaimer
Carlos D’Andrea Elimination Theory in the 21st century 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!
Some “Tapas” into the 21st century
Disclaimer These are not the highlights of the 21st century (so far), just a sampling of results
Carlos D’Andrea Elimination Theory in the 21st century , more will appear in the future... A lot of important names and interesting areas will be omitted...sorry!
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
Carlos D’Andrea Elimination Theory in the 21st century A lot of important names and interesting areas will be omitted...sorry!
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...
Carlos D’Andrea Elimination Theory in the 21st century sorry!
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... Carlos D’Andrea Elimination Theory in the 21st century 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! Carlos D’Andrea Elimination Theory in the 21st century The logarithmic limit set of a variety V ⊂ Cn is a fan in Rn, the Tropical Variety of V (MSC 2010 14Txx)
Tropical: from Exponential to Polynomial
Carlos D’Andrea Elimination Theory in the 21st century , the Tropical Variety of V (MSC 2010 14Txx)
Tropical: from Exponential to Polynomial
The logarithmic limit set of a variety V ⊂ Cn is a fan in Rn
Carlos D’Andrea Elimination Theory in the 21st century (MSC 2010 14Txx)
Tropical: from Exponential to Polynomial
The logarithmic limit set of a variety V ⊂ Cn is a fan in Rn, the Tropical Variety of V
Carlos D’Andrea Elimination Theory in the 21st century Tropical: from Exponential to Polynomial
The logarithmic limit set of a variety V ⊂ Cn is a fan in Rn, the Tropical Variety of V (MSC 2010 14Txx) Carlos D’Andrea Elimination Theory in the 21st century is “fast” encodes a lot of information about the original variety (dimension, degree, singularities...)
Computing Tropical Varieties
Carlos D’Andrea Elimination Theory in the 21st century encodes a lot of information about the original variety (dimension, degree, singularities...)
Computing Tropical Varieties
is “fast”
Carlos D’Andrea Elimination Theory in the 21st century (dimension, degree, singularities...)
Computing Tropical Varieties
is “fast” encodes a lot of information about the original variety
Carlos D’Andrea Elimination Theory in the 21st century Computing Tropical Varieties
is “fast” encodes a lot of information about the original variety (dimension, degree, singularities...)
Carlos D’Andrea Elimination Theory in the 21st century Alicia Dickenstein, Eva Maria Feitchner, Bernd Sturmfels “Tropical Discriminants” ( JAMS 2007)
First tapa: Tropical Discriminants
Carlos D’Andrea Elimination Theory in the 21st century First tapa: Tropical Discriminants
Alicia Dickenstein, Eva Maria Feitchner, Bernd Sturmfels “Tropical Discriminants” ( JAMS 2007)
Carlos D’Andrea Elimination Theory in the 21st century Theorem 1.1 For any d × n matrix A, the tropical A-discriminant τ(XA∗) equals the Minkowski sum of the co-Bergman fan B∗(A) and the row space of A.
Main Result
Carlos D’Andrea Elimination Theory in the 21st century For any d × n matrix A, the tropical A-discriminant τ(XA∗) equals the Minkowski sum of the co-Bergman fan B∗(A) and the row space of A.
Main Result
Theorem 1.1
Carlos D’Andrea Elimination Theory in the 21st century Main Result
Theorem 1.1 For any d × n matrix A, the tropical A-discriminant τ(XA∗) equals the Minkowski sum of the co-Bergman fan B∗(A) and the row space of A.
Carlos D’Andrea Elimination Theory in the 21st century “is” the tropicalization of the kernel of A
What is the co-Bergman fan?
Carlos D’Andrea Elimination Theory in the 21st century What is the co-Bergman fan?
“is” the tropicalization of the kernel of A
Carlos D’Andrea Elimination Theory in the 21st century Bernd Sturmfels, Jenia Tevelev “Elimination Theory for Tropical Varieties” ( MRL 2008)
Tapa # 2:Tropical Elimination
Carlos D’Andrea Elimination Theory in the 21st century Tapa # 2:Tropical Elimination
Bernd Sturmfels, Jenia Tevelev “Elimination Theory for Tropical Varieties” ( MRL 2008)
Carlos D’Andrea Elimination Theory in the 21st century Let α : Tn → Td be a homomorphism of tori and let A : Zn → Zd 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 )).
Theorem1.1
Carlos D’Andrea Elimination Theory in the 21st century and let A : Zn → Zd 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 )).
Theorem1.1
Let α : Tn → Td be a homomorphism of tori
Carlos D’Andrea Elimination Theory in the 21st century 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 )).
Theorem1.1
Let α : Tn → Td be a homomorphism of tori and let A : Zn → Zd be the corresponding linear map of lattices of one-parameter subgroups.
Carlos D’Andrea Elimination Theory in the 21st century Then A induces a generically finite map of tropical varieties from T (X ) onto T (α(X )).
Theorem1.1
Let α : Tn → Td be a homomorphism of tori and let A : Zn → Zd be the corresponding linear map of lattices of one-parameter subgroups. Suppose that α induces a generically finite morphism from X onto α(X ).
Carlos D’Andrea Elimination Theory in the 21st century Theorem1.1
Let α : Tn → Td be a homomorphism of tori and let A : Zn → Zd 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 )).
Carlos D’Andrea Elimination Theory in the 21st century Anders Jensen, Josephine Yu “Computing Tropical Resultants” (JA 2013)
What about Tropical Resultants?
Carlos D’Andrea Elimination Theory in the 21st century What about Tropical Resultants?
Anders Jensen, Josephine Yu “Computing Tropical Resultants” (JA 2013)
Carlos D’Andrea Elimination Theory in the 21st century We fix the supports A = (A1,..., Ak) 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.
From the abstract:
Carlos D’Andrea Elimination Theory in the 21st century From the abstract:
We fix the supports A = (A1,..., Ak) 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.
Carlos D’Andrea Elimination Theory in the 21st century 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...
From the abstract II
Carlos D’Andrea Elimination Theory in the 21st century 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...
Carlos D’Andrea Elimination Theory in the 21st century 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 codimension1...
From the abstract III
Carlos D’Andrea Elimination Theory in the 21st century 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 codimension1...
From the abstract III
Carlos D’Andrea Elimination Theory in the 21st century 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 codimension1...
Carlos D’Andrea Elimination Theory in the 21st century In the “real world” systems of equations are neither homogeneous nor all the monomials appear in the expansion 2 2 3 F0 = a01 + a02X1 X2 + a03X1 X2 2 2 F1 = a10 + a11X1 + a12X1 X2 3 F2 = a20X1 + a21X1 X2
“Relaxed” resultants
Carlos D’Andrea Elimination Theory in the 21st century “Relaxed” resultants In the “real world” systems of equations are neither homogeneous nor all the monomials appear in the expansion 2 2 3 F0 = a01 + a02X1 X2 + a03X1 X2 2 2 F1 = a10 + a11X1 + a12X1 X2 3 F2 = a20X1 + a21X1 X2
Carlos D’Andrea Elimination Theory in the 21st century D, Mart´ınSombra “A Poisson formula for the sparse resultant” (PLMS 2015)
A0 An × n W = {(ci,a, ξ): Fi (ξ) = 0 ∀i} ⊂ P × ... × P × (C ) ↓ ↓ π π(W ) ⊂ PA0 × ... × PAn
The sparse resultant of F0,..., Fn is the defining equation of the direct image π∗W n d 7→ MV (N(A1),..., N(An)) n−1 nd L 7→ nMV (N(A1),..., N(An))L
Sparse Resultants (after GKZ)
Carlos D’Andrea Elimination Theory in the 21st century The sparse resultant of F0,..., Fn is the defining equation of the direct image π∗W n d 7→ MV (N(A1),..., N(An)) n−1 nd L 7→ nMV (N(A1),..., N(An))L
Sparse Resultants (after GKZ)
D, Mart´ınSombra “A Poisson formula for the sparse resultant” (PLMS 2015)
A0 An × n W = {(ci,a, ξ): Fi (ξ) = 0 ∀i} ⊂ P × ... × P × (C ) ↓ ↓ π π(W ) ⊂ PA0 × ... × PAn
Carlos D’Andrea Elimination Theory in the 21st century n d 7→ MV (N(A1),..., N(An)) n−1 nd L 7→ nMV (N(A1),..., N(An))L
Sparse Resultants (after GKZ)
D, Mart´ınSombra “A Poisson formula for the sparse resultant” (PLMS 2015)
A0 An × n W = {(ci,a, ξ): Fi (ξ) = 0 ∀i} ⊂ P × ... × P × (C ) ↓ ↓ π π(W ) ⊂ PA0 × ... × PAn
The sparse resultant of F0,..., Fn is the defining equation of the direct image π∗W
Carlos D’Andrea Elimination Theory in the 21st century n−1 nd L 7→ nMV (N(A1),..., N(An))L
Sparse Resultants (after GKZ)
D, Mart´ınSombra “A Poisson formula for the sparse resultant” (PLMS 2015)
A0 An × n W = {(ci,a, ξ): Fi (ξ) = 0 ∀i} ⊂ P × ... × P × (C ) ↓ ↓ π π(W ) ⊂ PA0 × ... × PAn
The sparse resultant of F0,..., Fn is the defining equation of the direct image π∗W n d 7→ MV (N(A1),..., N(An))
Carlos D’Andrea Elimination Theory in the 21st century Sparse Resultants (after GKZ)
D, Mart´ınSombra “A Poisson formula for the sparse resultant” (PLMS 2015)
A0 An × n W = {(ci,a, ξ): Fi (ξ) = 0 ∀i} ⊂ P × ... × P × (C ) ↓ ↓ π π(W ) ⊂ PA0 × ... × PAn
The sparse resultant of F0,..., Fn is the defining equation of the direct image π∗W n d 7→ MV (N(A1),..., N(An)) n−1 nd L 7→ nMV (N(A1),..., N(An))L
Carlos D’Andrea Elimination Theory in the 21st century By replacing( C×)n with other “efficient” varieties X , we get “efficient” resultants:
N0 Nn W = {(ci,a, ξ): Fi (ξ) = 0 ∀i} ⊂ P × ... × P × X ↓ ↓ π π(W ) ⊂ PN0 × ... × PNn
Other “relaxed” resultants
Carlos D’Andrea Elimination Theory in the 21st century Other “relaxed” resultants
By replacing( C×)n with other “efficient” varieties X , we get “efficient” resultants:
N0 Nn W = {(ci,a, ξ): Fi (ξ) = 0 ∀i} ⊂ P × ... × P × X ↓ ↓ π π(W ) ⊂ PN0 × ... × PNn
Carlos D’Andrea Elimination Theory in the 21st century Weighted Resultants (Jouanolou) Parametric Resultants (Bus´e-Elkadi-Mourrain) Residual Resultants (Bus´e) ...
Examples of “relaxed” resultants
Reduced Resultants (Zariski, Jouanolou,...)
Carlos D’Andrea Elimination Theory in the 21st century Parametric Resultants (Bus´e-Elkadi-Mourrain) Residual Resultants (Bus´e) ...
Examples of “relaxed” resultants
Reduced Resultants (Zariski, Jouanolou,...) Weighted Resultants (Jouanolou)
Carlos D’Andrea Elimination Theory in the 21st century Residual Resultants (Bus´e) ...
Examples of “relaxed” resultants
Reduced Resultants (Zariski, Jouanolou,...) Weighted Resultants (Jouanolou) Parametric Resultants (Bus´e-Elkadi-Mourrain)
Carlos D’Andrea Elimination Theory in the 21st century ...
Examples of “relaxed” resultants
Reduced Resultants (Zariski, Jouanolou,...) Weighted Resultants (Jouanolou) Parametric Resultants (Bus´e-Elkadi-Mourrain) Residual Resultants (Bus´e)
Carlos D’Andrea Elimination Theory in the 21st century Examples of “relaxed” resultants
Reduced Resultants (Zariski, Jouanolou,...) Weighted Resultants (Jouanolou) Parametric Resultants (Bus´e-Elkadi-Mourrain) Residual Resultants (Bus´e) ...
Carlos D’Andrea Elimination Theory in the 21st century Recall van der Waerden’s matrix introduced this morning: . . . . β α Ms(λ) x = x Fi . . . .
For s 0, the rank of Ms(λ) drops iff there is a common solution
Implicitization matrices
Carlos D’Andrea Elimination Theory in the 21st century For s 0, the rank of Ms(λ) drops iff there is a common solution
Implicitization matrices
Recall van der Waerden’s matrix introduced this morning: . . . . β α Ms(λ) x = x Fi . . . .
Carlos D’Andrea Elimination Theory in the 21st century Implicitization matrices
Recall van der Waerden’s matrix introduced this morning: . . . . β α Ms(λ) x = x Fi . . . .
For s 0, the rank of Ms(λ) drops iff there is a common solution Carlos D’Andrea Elimination Theory in the 21st century Laurent Bus´e “Implicit matrix representations of rational B´eziercurves and surfaces” (CAGD 2014)
“Rank” vs “Determinant”
Carlos D’Andrea Elimination Theory in the 21st century “Rank” vs “Determinant”
Laurent Bus´e “Implicit matrix representations of rational B´eziercurves and surfaces” (CAGD 2014)
Carlos D’Andrea Elimination Theory in the 21st century Classic implicitization: Compute the equations Fi (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
Idea
Given a rational parametrization of a spatial curve( φ1(t), φ2(t), φ3(t))
Carlos D’Andrea Elimination Theory in the 21st century Compute the equations Fi (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
Idea
Given a rational parametrization of a spatial curve( φ1(t), φ2(t), φ3(t)) Classic implicitization:
Carlos D’Andrea Elimination Theory in the 21st century 21st century implicitization: Compute a matrix M(X , Y , Z) such that its rank drops on the points of the curve
Idea
Given a rational parametrization of a spatial curve( φ1(t), φ2(t), φ3(t)) Classic implicitization: Compute the equations Fi (X , Y , Z) of the image
Carlos D’Andrea Elimination Theory in the 21st century Compute a matrix M(X , Y , Z) such that its rank drops on the points of the curve
Idea
Given a rational parametrization of a spatial curve( φ1(t), φ2(t), φ3(t)) Classic implicitization: Compute the equations Fi (X , Y , Z) of the image 21st century implicitization:
Carlos D’Andrea Elimination Theory in the 21st century Idea
Given a rational parametrization of a spatial curve( φ1(t), φ2(t), φ3(t)) Classic implicitization: Compute the equations Fi (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
Carlos D’Andrea Elimination Theory in the 21st century 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,...)
Properties
Carlos D’Andrea Elimination Theory in the 21st century 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,...)
Properties
Entries are linear in X , Y , Z
Carlos D’Andrea Elimination Theory in the 21st century Over the reals one can use SVD and other numerical methods Well suited for a lot of problems in CAGD (properness, inversion,...)
Properties
Entries are linear in X , Y , Z Testing rank is fast by using numerical methods
Carlos D’Andrea Elimination Theory in the 21st century Well suited for a lot of problems in CAGD (properness, inversion,...)
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
Carlos D’Andrea Elimination Theory in the 21st century 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,...)
Carlos D’Andrea Elimination Theory in the 21st century More about this on Wednesday!
Carlos D’Andrea Elimination Theory in the 21st century Warming up tapa
The rest on Wednesday :-)
Syzygies and Rees Algebras
Carlos D’Andrea Elimination Theory in the 21st century The rest on Wednesday :-)
Syzygies and Rees Algebras
Warming up tapa
Carlos D’Andrea Elimination Theory in the 21st century Syzygies and Rees Algebras
Warming up tapa
The rest on Wednesday :-)
Carlos D’Andrea Elimination Theory in the 21st century The implicit equation of a rational quartic can be computed as a2 × 2 determinant. If the curve has a triple point, then one row is linear and the other is cubic. Otherwise, both rows are quadratic.
Example (Sederberg & Chen 1995)
Carlos D’Andrea Elimination Theory in the 21st century If the curve has a triple point, then one row is linear and the other is cubic. Otherwise, both rows are quadratic.
Example (Sederberg & Chen 1995)
The implicit equation of a rational quartic can be computed as a2 × 2 determinant.
Carlos D’Andrea Elimination Theory in the 21st century Otherwise, both rows are quadratic.
Example (Sederberg & Chen 1995)
The implicit equation of a rational quartic can be computed as a2 × 2 determinant. If the curve has a triple point, then one row is linear and the other is cubic.
Carlos D’Andrea Elimination Theory in the 21st century Example (Sederberg & Chen 1995)
The implicit equation of a rational quartic can be computed as a2 × 2 determinant. If the curve has a triple point, then one row is linear and the other is cubic. Otherwise, both rows are quadratic.
Carlos D’Andrea Elimination Theory in the 21st century 4 4 2 2 3 φ(t0, t1) = (t0 − t1 : −t0 t1 : t0t1 )
4 4 2 F (X0, X1, X2) = X2 − X1 − X0X1X2 L1,1(T , X ) = T0X2 + T1X1 3 2 3 L1,3(T , X ) = T0(X1 + X0X2 ) + T1 X2 X2 X1 3 2 3 X1 + X0X2 X2
A quartic with a triple point
Carlos D’Andrea Elimination Theory in the 21st century 4 4 2 F (X0, X1, X2) = X2 − X1 − X0X1X2 L1,1(T , X ) = T0X2 + T1X1 3 2 3 L1,3(T , X ) = T0(X1 + X0X2 ) + T1 X2 X2 X1 3 2 3 X1 + X0X2 X2
A quartic with a triple point
4 4 2 2 3 φ(t0, t1) = (t0 − t1 : −t0 t1 : t0t1 )
Carlos D’Andrea Elimination Theory in the 21st century L1,1(T , X ) = T0X2 + T1X1 3 2 3 L1,3(T , X ) = T0(X1 + X0X2 ) + T1 X2 X2 X1 3 2 3 X1 + X0X2 X2
A quartic with a triple point
4 4 2 2 3 φ(t0, t1) = (t0 − t1 : −t0 t1 : t0t1 )
4 4 2 F (X0, X1, X2) = X2 − X1 − X0X1X2
Carlos D’Andrea Elimination Theory in the 21st century A quartic with a triple point
4 4 2 2 3 φ(t0, t1) = (t0 − t1 : −t0 t1 : t0t1 )
4 4 2 F (X0, X1, X2) = X2 − X1 − X0X1X2 L1,1(T , X ) = T0X2 + T1X1 3 2 3 L1,3(T , X ) = T0(X1 + X0X2 ) + T1 X2 X2 X1 3 2 3 X1 + X0X2 X2 Carlos D’Andrea Elimination Theory in the 21st century 4 3 2 2 2 2 2 3 F (X ) = X2 +4X0X1 +2X0X1X2 −16X0 X1 −6X0 X2 +16X0 X1 2 2 L1,2(T , X ) = T0(X1X2 − X0X2) + T1(−X2 − 2X0X1 + 4X0 ) ˜ 2 1 2 L1,2(T , X ) = T0(X1 + 2 X2 − 2X0X1) + T1(X0X2 − X1X2)
A quartic without triple points
4 2 2 4 3 3 φ(t0 : t1) = (t0 : 6t0 t1 − 4t1 : 4t0 t1 − 4t0t1 )
Carlos D’Andrea Elimination Theory in the 21st century 2 2 L1,2(T , X ) = T0(X1X2 − X0X2) + T1(−X2 − 2X0X1 + 4X0 ) ˜ 2 1 2 L1,2(T , X ) = T0(X1 + 2 X2 − 2X0X1) + T1(X0X2 − X1X2)
A quartic without triple points
4 2 2 4 3 3 φ(t0 : t1) = (t0 : 6t0 t1 − 4t1 : 4t0 t1 − 4t0t1 )
4 3 2 2 2 2 2 3 F (X ) = X2 +4X0X1 +2X0X1X2 −16X0 X1 −6X0 X2 +16X0 X1
Carlos D’Andrea Elimination Theory in the 21st century A quartic without triple points
4 2 2 4 3 3 φ(t0 : t1) = (t0 : 6t0 t1 − 4t1 : 4t0 t1 − 4t0t1 )
4 3 2 2 2 2 2 3 F (X ) = X2 +4X0X1 +2X0X1X2 −16X0 X1 −6X0 X2 +16X0 X1 2 2 L1,2(T , X ) = T0(X1X2 − X0X2) + T1(−X2 − 2X0X1 + 4X0 ) ˜ 2 1 2 L1,2(T , X ) = T0(X1 + 2 X2 − 2X0X1) + T1(X0X2 − X1X2)
Carlos D’Andrea Elimination Theory in the 21st century If the curve has a point of multiplicity d − 1 the implicit equation is always a2 × 2 determinant of a moving line and a moving curve of degree d − 1 0 L1,1(X ) L1,1(X ) 0 L1,d−1(X ) L1,d−1(X )
A very compact formula
Carlos D’Andrea Elimination Theory in the 21st century the implicit equation is always a2 × 2 determinant of a moving line and a moving curve of degree d − 1 0 L1,1(X ) L1,1(X ) 0 L1,d−1(X ) L1,d−1(X )
A very compact formula
If the curve has a point of multiplicity d − 1
Carlos D’Andrea Elimination Theory in the 21st century A very compact formula
If the curve has a point of multiplicity d − 1 the implicit equation is always a2 × 2 determinant of a moving line and a moving curve of degree d − 1 0 L1,1(X ) L1,1(X ) 0 L1,d−1(X ) L1,d−1(X )
Carlos D’Andrea Elimination Theory in the 21st century 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
The “method” of moving curves
Carlos D’Andrea Elimination Theory in the 21st century
some moving lines
some moving conics
some moving cubics
··· the more singular the curve, the simpler the description of the determinant
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
Carlos D’Andrea Elimination Theory in the 21st century the more singular the curve, the simpler the description of the determinant
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
···
Carlos D’Andrea Elimination Theory in the 21st century 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
Carlos D’Andrea Elimination Theory in the 21st century
which moving lines?
which moving conics?
which moving cubics?
···
In general, we do not know..
Carlos D’Andrea Elimination Theory in the 21st century In general, we do not know..
which moving lines?
which moving conics?
which moving cubics?
···
Carlos D’Andrea Elimination Theory in the 21st century David Cox “The moving curve ideal and the Rees algebra” (TCS 2008) Kφ := {Moving curves following φ} = homogeneous elements in the kernel of
K[T0, T1, X0, X1, X2] → K[T0, T1, s] Ti 7→ Ti X0 7→ φ0(T )s X1 7→ φ1(T )s X2 7→ φ2(T )s The image of this map is the Rees Algebra of φ
Some order in the 21st century
Carlos D’Andrea Elimination Theory in the 21st century Kφ := {Moving curves following φ} = homogeneous elements in the kernel of
K[T0, T1, X0, X1, X2] → K[T0, T1, s] Ti 7→ Ti X0 7→ φ0(T )s X1 7→ φ1(T )s X2 7→ φ2(T )s The image of this map is the Rees Algebra of φ
Some order in the 21st century
David Cox “The moving curve ideal and the Rees algebra” (TCS 2008)
Carlos D’Andrea Elimination Theory in the 21st century The image of this map is the Rees Algebra of φ
Some order in the 21st century
David Cox “The moving curve ideal and the Rees algebra” (TCS 2008) Kφ := {Moving curves following φ} = homogeneous elements in the kernel of
K[T0, T1, X0, X1, X2] → K[T0, T1, s] Ti 7→ Ti X0 7→ φ0(T )s X1 7→ φ1(T )s X2 7→ φ2(T )s
Carlos D’Andrea Elimination Theory in the 21st century Some order in the 21st century
David Cox “The moving curve ideal and the Rees algebra” (TCS 2008) Kφ := {Moving curves following φ} = homogeneous elements in the kernel of
K[T0, T1, X0, X1, X2] → K[T0, T1, s] Ti 7→ Ti X0 7→ φ0(T )s X1 7→ φ1(T )s X2 7→ φ2(T )s The image of this map is the Rees Algebra of φ
Carlos D’Andrea Elimination Theory in the 21st century The implicit equation should be obtained as the determinant of a matrix with
···
some minimal generators of Kφ
and relations among them
···
The more singular the curve, the “simpler” the description of Kφ
Method of moving curves revisited
Carlos D’Andrea Elimination Theory in the 21st century
···
some minimal generators of Kφ
and relations among them
···
The more singular the curve, the “simpler” the description of Kφ
Method of moving curves revisited
The implicit equation should be obtained as the determinant of a matrix with
Carlos D’Andrea Elimination Theory in the 21st century The more singular the curve, the “simpler” the description of Kφ
Method of moving curves revisited
The implicit equation should be obtained as the determinant of a matrix with
···
some minimal generators of Kφ
and relations among them
···
Carlos D’Andrea Elimination Theory in the 21st century Method of moving curves revisited
The implicit equation should be obtained as the determinant of a matrix with
···
some minimal generators of Kφ
and relations among them
···
The more singular the curve, the “simpler” the description of Kφ
Carlos D’Andrea Elimination Theory in the 21st century Compute a minimal system of generators of Kφ for any φ
We are working on that! (More on Wednesday)
21st Century Problem
Carlos D’Andrea Elimination Theory in the 21st century for any φ
We are working on that! (More on Wednesday)
21st Century Problem
Compute a minimal system of generators of Kφ
Carlos D’Andrea Elimination Theory in the 21st century We are working on that! (More on Wednesday)
21st Century Problem
Compute a minimal system of generators of Kφ for any φ
Carlos D’Andrea Elimination Theory in the 21st century (More on Wednesday)
21st Century Problem
Compute a minimal system of generators of Kφ for any φ
We are working on that!
Carlos D’Andrea Elimination Theory in the 21st century 21st Century Problem
Compute a minimal system of generators of Kφ for any φ
We are working on that! (More on Wednesday)
Carlos D’Andrea Elimination Theory in the 21st century 21st Century Problem
Compute a minimal system of generators of Kφ for any φ
We are working on that! (More on Wednesday)
Carlos D’Andrea Elimination Theory in the 21st century Changing the model of computation
Carlos D’Andrea Elimination Theory in the 21st century Gabriela Jeronimo, Teresa Krick, Juan Sabia, Mart´ınSombra “The Computational Complexity of the Chow Form” ( JFoCM 2004)
Probabilistic Algorithms: faster they are
Carlos D’Andrea Elimination Theory in the 21st century Probabilistic Algorithms: faster they are
Gabriela Jeronimo, Teresa Krick, Juan Sabia, Mart´ınSombra “The Computational Complexity of the Chow Form” ( JFoCM 2004)
Carlos D’Andrea Elimination Theory in the 21st century 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...
From the abstract:
We present a bounded probability algorithm for the computation of the Chow forms (resultants) of the equidimensional components of a variety...
Carlos D’Andrea Elimination Theory in the 21st century 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...
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...
Carlos D’Andrea Elimination Theory in the 21st century 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...
Carlos D’Andrea Elimination Theory in the 21st century Given (equations) of a d-dimensional variety V ⊂ Pn Cut V with “general” (probabilistically) d hyperplanes to obtain finite points: V ∩ {x1 = ... = xd = 0} = {ξ1, . . . , ξD}
From (a geometric resolution of) {ξ1, . . . , ξD}, lift (Newton’s Method) to the Chow Form of V ∩ {x1 = ... = xd−1 = 0} Do it( d − 1) times more...
Main idea
Carlos D’Andrea Elimination Theory in the 21st century Cut V with “general” (probabilistically) d hyperplanes to obtain finite points: V ∩ {x1 = ... = xd = 0} = {ξ1, . . . , ξD}
From (a geometric resolution of) {ξ1, . . . , ξD}, lift (Newton’s Method) to the Chow Form of V ∩ {x1 = ... = xd−1 = 0} Do it( d − 1) times more...
Main idea
Given (equations) of a d-dimensional variety V ⊂ Pn
Carlos D’Andrea Elimination Theory in the 21st century Do it( d − 1) times more...
Main idea
Given (equations) of a d-dimensional variety V ⊂ Pn Cut V with “general” (probabilistically) d hyperplanes to obtain finite points: V ∩ {x1 = ... = xd = 0} = {ξ1, . . . , ξD}
From (a geometric resolution of) {ξ1, . . . , ξD}, lift (Newton’s Method) to the Chow Form of V ∩ {x1 = ... = xd−1 = 0}
Carlos D’Andrea Elimination Theory in the 21st century Main idea
Given (equations) of a d-dimensional variety V ⊂ Pn Cut V with “general” (probabilistically) d hyperplanes to obtain finite points: V ∩ {x1 = ... = xd = 0} = {ξ1, . . . , ξD}
From (a geometric resolution of) {ξ1, . . . , ξD}, lift (Newton’s Method) to the Chow Form of V ∩ {x1 = ... = xd−1 = 0} Do it( d − 1) times more...
Carlos D’Andrea Elimination Theory in the 21st century Gabriela Jeronimo, Juan Sabia “Sparse resultants and straight-line programs” (JSC 2018)
Star Model: Straight-Line Programs
Carlos D’Andrea Elimination Theory in the 21st century Star Model: Straight-Line Programs
Gabriela Jeronimo, Juan Sabia “Sparse resultants and straight-line programs” (JSC 2018)
Carlos D’Andrea Elimination Theory in the 21st century It is a program with no branches, no loops, no conditional statements, no comparisons...just a sequence of basic operations
What is a straight-line program?
Carlos D’Andrea Elimination Theory in the 21st century , no loops, no conditional statements, no comparisons...just a sequence of basic operations
What is a straight-line program?
It is a program with no branches
Carlos D’Andrea Elimination Theory in the 21st century , no conditional statements, no comparisons...just a sequence of basic operations
What is a straight-line program?
It is a program with no branches, no loops
Carlos D’Andrea Elimination Theory in the 21st century , no comparisons...just a sequence of basic operations
What is a straight-line program?
It is a program with no branches, no loops, no conditional statements
Carlos D’Andrea Elimination Theory in the 21st century ...just a sequence of basic operations
What is a straight-line program?
It is a program with no branches, no loops, no conditional statements, no comparisons
Carlos D’Andrea Elimination Theory in the 21st century 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 Carlos D’Andrea Elimination Theory in the 21st century 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.
From the abstract:
Carlos D’Andrea Elimination Theory in the 21st century 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.
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 ...
Carlos D’Andrea Elimination Theory in the 21st century 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.
Carlos D’Andrea Elimination Theory in the 21st century SLP’s Geometric Resolution Newton Hensel Lifting Pad´eApproximations...
Main ideas
Carlos D’Andrea Elimination Theory in the 21st century Geometric Resolution Newton Hensel Lifting Pad´eApproximations...
Main ideas
SLP’s
Carlos D’Andrea Elimination Theory in the 21st century Newton Hensel Lifting Pad´eApproximations...
Main ideas
SLP’s Geometric Resolution
Carlos D’Andrea Elimination Theory in the 21st century Pad´eApproximations...
Main ideas
SLP’s Geometric Resolution Newton Hensel Lifting
Carlos D’Andrea Elimination Theory in the 21st century Main ideas
SLP’s Geometric Resolution Newton Hensel Lifting Pad´eApproximations...
Carlos D’Andrea Elimination Theory in the 21st century 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
Anoter “real” model: BSS-machine
Carlos D’Andrea Elimination Theory in the 21st century with registers that can store arbitrary real numbers and that can compute rational functions over the reals at unit cost
Anoter “real” model: BSS-machine
is a Random Access Machine
Carlos D’Andrea Elimination Theory in the 21st century and that can compute rational functions over the reals at unit cost
Anoter “real” model: BSS-machine
is a Random Access Machine with registers that can store arbitrary real numbers
Carlos D’Andrea Elimination Theory in the 21st century 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
Carlos D’Andrea Elimination Theory in the 21st century Given f (x) ∈ Z[x], compute a small approximate root, fast
Parallelize: find just one solution
Carlos D’Andrea Elimination Theory in the 21st century approximate root, fast
Parallelize: find just one solution
Given f (x) ∈ Z[x], compute a small
Carlos D’Andrea Elimination Theory in the 21st century , fast
Parallelize: find just one solution
Given f (x) ∈ Z[x], compute a small approximate root
Carlos D’Andrea Elimination Theory in the 21st century Parallelize: find just one solution
Given f (x) ∈ Z[x], compute a small approximate root, fast
Carlos D’Andrea Elimination Theory in the 21st century What is an approximate root?
Carlos D’Andrea Elimination Theory in the 21st century 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 ?
Steve Smale’s 17th’s problem (1998)
Carlos D’Andrea Elimination Theory in the 21st century 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 ? Carlos D’Andrea Elimination Theory in the 21st century Carlos Beltr´an, Luis Miguel Pardo. “On Smale’s 17th Problem: A Probabilistic Positive answer” ( JFoCM 2008) Carlos Beltr´an, Luis Miguel Pardo. “Smale’s 17th Problem: Average Polynomial Time to compute affine and projective solutions” (JAMS 2009) Felipe Cucker, Peter B¨urgisser. “On a problem posed by Steve Smale” (Ann.Math 2011)
Solving Smale’s 17th Problem
Carlos D’Andrea Elimination Theory in the 21st century Carlos Beltr´an, Luis Miguel Pardo. “Smale’s 17th Problem: Average Polynomial Time to compute affine and projective solutions” (JAMS 2009) Felipe Cucker, Peter B¨urgisser. “On a problem posed by Steve Smale” (Ann.Math 2011)
Solving Smale’s 17th Problem
Carlos Beltr´an, Luis Miguel Pardo. “On Smale’s 17th Problem: A Probabilistic Positive answer” ( JFoCM 2008)
Carlos D’Andrea Elimination Theory in the 21st century Felipe Cucker, Peter B¨urgisser. “On a problem posed by Steve Smale” (Ann.Math 2011)
Solving Smale’s 17th Problem
Carlos Beltr´an, Luis Miguel Pardo. “On Smale’s 17th Problem: A Probabilistic Positive answer” ( JFoCM 2008) Carlos Beltr´an, Luis Miguel Pardo. “Smale’s 17th Problem: Average Polynomial Time to compute affine and projective solutions” (JAMS 2009)
Carlos D’Andrea Elimination Theory in the 21st century Solving Smale’s 17th Problem
Carlos Beltr´an, Luis Miguel Pardo. “On Smale’s 17th Problem: A Probabilistic Positive answer” ( JFoCM 2008) Carlos Beltr´an, Luis Miguel Pardo. “Smale’s 17th Problem: Average Polynomial Time to compute affine and projective solutions” (JAMS 2009) Felipe Cucker, Peter B¨urgisser. “On a problem posed by Steve Smale” (Ann.Math 2011)
Carlos D’Andrea Elimination Theory in the 21st century Solving Smale’s 17th Problem
Carlos Beltr´an, Luis Miguel Pardo. “On Smale’s 17th Problem: A Probabilistic Positive answer” ( JFoCM 2008) Carlos Beltr´an, Luis Miguel Pardo. “Smale’s 17th Problem: Average Polynomial Time to compute affine and projective solutions” (JAMS 2009) Felipe Cucker, Peter B¨urgisser. “On a problem posed by Steve Smale” (Ann.Math 2011)
Carlos D’Andrea Elimination Theory in the 21st century “A deterministic algorithm to compute approximate roots of polynomial systems in polynomial average time” JFoCM (2017)
Pierre Lairez (2017)
Carlos D’Andrea Elimination Theory in the 21st century “A deterministic algorithm to compute approximate roots of polynomial systems in polynomial average time” JFoCM (2017)
Pierre Lairez (2017)
Carlos D’Andrea Elimination Theory in the 21st century Pierre Lairez (2017)
“A deterministic algorithm to compute approximate roots of polynomial systems in polynomial average time” JFoCM (2017)
Carlos D’Andrea Elimination Theory in the 21st century Start with an “easy” system “Chase” the roots with an homotopy + Newton’s method
Homotopies (In depth tomorrow!)
Carlos D’Andrea Elimination Theory in the 21st century “Chase” the roots with an homotopy + Newton’s method
Homotopies (In depth tomorrow!)
Start with an “easy” system
Carlos D’Andrea Elimination Theory in the 21st century Homotopies (In depth tomorrow!)
Start with an “easy” system “Chase” the roots with an homotopy + Newton’s method
Carlos D’Andrea Elimination Theory in the 21st century By using homotopies, you can compute very fast Components Multiplicities Decomposition Dimension ...
Numerical Algebraic Geometry
Carlos D’Andrea Elimination Theory in the 21st century Components Multiplicities Decomposition Dimension ...
Numerical Algebraic Geometry
By using homotopies, you can compute very fast
Carlos D’Andrea Elimination Theory in the 21st century Multiplicities Decomposition Dimension ...
Numerical Algebraic Geometry
By using homotopies, you can compute very fast Components
Carlos D’Andrea Elimination Theory in the 21st century Decomposition Dimension ...
Numerical Algebraic Geometry
By using homotopies, you can compute very fast Components Multiplicities
Carlos D’Andrea Elimination Theory in the 21st century Dimension ...
Numerical Algebraic Geometry
By using homotopies, you can compute very fast Components Multiplicities Decomposition
Carlos D’Andrea Elimination Theory in the 21st century ...
Numerical Algebraic Geometry
By using homotopies, you can compute very fast Components Multiplicities Decomposition Dimension
Carlos D’Andrea Elimination Theory in the 21st century Numerical Algebraic Geometry
By using homotopies, you can compute very fast Components Multiplicities Decomposition Dimension ...
Carlos D’Andrea Elimination Theory in the 21st century Differential/Difference Elimination Real Elimination Applications Implementations ...
Not in the menu today...
Carlos D’Andrea Elimination Theory in the 21st century Real Elimination Applications Implementations ...
Not in the menu today...
Differential/Difference Elimination
Carlos D’Andrea Elimination Theory in the 21st century Applications Implementations ...
Not in the menu today...
Differential/Difference Elimination Real Elimination
Carlos D’Andrea Elimination Theory in the 21st century Implementations ...
Not in the menu today...
Differential/Difference Elimination Real Elimination Applications
Carlos D’Andrea Elimination Theory in the 21st century ...
Not in the menu today...
Differential/Difference Elimination Real Elimination Applications Implementations
Carlos D’Andrea Elimination Theory in the 21st century Not in the menu today...
Differential/Difference Elimination Real Elimination Applications Implementations ... Carlos D’Andrea Elimination Theory in the 21st century The 21st century is/will be driven by Applications Fast & efficient computations Interesting challenges and problems need your help here!
Conclusion
Carlos D’Andrea Elimination Theory in the 21st century Fast & efficient computations Interesting challenges and problems need your help here!
Conclusion
The 21st century is/will be driven by Applications
Carlos D’Andrea Elimination Theory in the 21st century Interesting challenges and problems need your help here!
Conclusion
The 21st century is/will be driven by Applications Fast & efficient computations
Carlos D’Andrea Elimination Theory in the 21st century Conclusion
The 21st century is/will be driven by Applications Fast & efficient computations Interesting challenges and problems need your help here!
Carlos D’Andrea Elimination Theory in the 21st century Thanks!
http://www.ub.edu/arcades/cdandrea.html
Carlos D’Andrea Elimination Theory in the 21st century