Discrete geometry from an algebraic point of view · Discrete geometry from an algebraic point of...
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Discrete geometry from an algebraic point of view
Valculescu Claudiu
4 September 2014
Valculescu Claudiu Discrete geometry from an algebraic point of view 1 / 25
Preliminaries
DefinitionDiscrete geometry is concerned with the study of finite (discrete) setsof geometric objects and their properties.
Some topics:Packings, coverings and tilingsFor example: What is the maximum density of a packing of unitcircles in the plane?
Valculescu Claudiu Discrete geometry from an algebraic point of view 2 / 25
Preliminaries
DefinitionDiscrete geometry is concerned with the study of finite (discrete) setsof geometric objects and their properties.
Some topics:Packings, coverings and tilings
For example: What is the maximum density of a packing of unitcircles in the plane?
Valculescu Claudiu Discrete geometry from an algebraic point of view 2 / 25
Preliminaries
DefinitionDiscrete geometry is concerned with the study of finite (discrete) setsof geometric objects and their properties.
Some topics:Packings, coverings and tilingsFor example: What is the maximum density of a packing of unitcircles in the plane?
Valculescu Claudiu Discrete geometry from an algebraic point of view 2 / 25
Preliminaries
DefinitionDiscrete geometry is concerned with the study of finite (discrete) setsof geometric objects and their properties.
Some topics:Packings, coverings and tilingsFor example: What is the maximum density of a packing of unitcircles in the plane?
Valculescu Claudiu Discrete geometry from an algebraic point of view 2 / 25
PreliminariesSome topics:
Incidence structuresFor example: Given an arbitrary set P of n points and an arbitraryset L of m lines, both in the real plane, what is the maximum numberof point-line incidences, i.e. the maximum cardinality of the set
I(P, L) = (p, l) : p ∈ l , p ∈ P, l ∈ L?
l1
l3
l2l4p1
p4p2
p3
(p4, l1)(p3, l2)
(p4, l2)
(p1, l3)(p1, l4)(p2, l4)
In particular, we are interested in the asymptotic behaviour asm, n→∞.
Valculescu Claudiu Discrete geometry from an algebraic point of view 3 / 25
PreliminariesSome topics:
Incidence structures
For example: Given an arbitrary set P of n points and an arbitraryset L of m lines, both in the real plane, what is the maximum numberof point-line incidences, i.e. the maximum cardinality of the set
I(P, L) = (p, l) : p ∈ l , p ∈ P, l ∈ L?
l1
l3
l2l4p1
p4p2
p3
(p4, l1)(p3, l2)
(p4, l2)
(p1, l3)(p1, l4)(p2, l4)
In particular, we are interested in the asymptotic behaviour asm, n→∞.
Valculescu Claudiu Discrete geometry from an algebraic point of view 3 / 25
PreliminariesSome topics:
Incidence structuresFor example: Given an arbitrary set P of n points and an arbitraryset L of m lines, both in the real plane, what is the maximum numberof point-line incidences, i.e. the maximum cardinality of the set
I(P, L) = (p, l) : p ∈ l , p ∈ P, l ∈ L?
l1
l3
l2l4p1
p4p2
p3
(p4, l1)(p3, l2)
(p4, l2)
(p1, l3)(p1, l4)(p2, l4)
In particular, we are interested in the asymptotic behaviour asm, n→∞.
Valculescu Claudiu Discrete geometry from an algebraic point of view 3 / 25
PreliminariesSome topics:
Incidence structuresFor example: Given an arbitrary set P of n points and an arbitraryset L of m lines, both in the real plane, what is the maximum numberof point-line incidences, i.e. the maximum cardinality of the set
I(P, L) = (p, l) : p ∈ l , p ∈ P, l ∈ L?
l1
l3
l2l4p1
p4p2
p3
(p4, l1)(p3, l2)
(p4, l2)
(p1, l3)(p1, l4)(p2, l4)
In particular, we are interested in the asymptotic behaviour asm, n→∞.
Valculescu Claudiu Discrete geometry from an algebraic point of view 3 / 25
PreliminariesSome topics:
Incidence structuresFor example: Given an arbitrary set P of n points and an arbitraryset L of m lines, both in the real plane, what is the maximum numberof point-line incidences, i.e. the maximum cardinality of the set
I(P, L) = (p, l) : p ∈ l , p ∈ P, l ∈ L?
l1
l3
l2l4p1
p4p2
p3
(p4, l1)(p3, l2)
(p4, l2)
(p1, l3)(p1, l4)(p2, l4)
In particular, we are interested in the asymptotic behaviour asm, n→∞.
Valculescu Claudiu Discrete geometry from an algebraic point of view 3 / 25
PreliminariesSome other topics:
Geometric graph theoryFor example: Planarity, crossing number, ...
Topological combinatoricsFor example: Fair division problems, equipartitions
Valculescu Claudiu Discrete geometry from an algebraic point of view 4 / 25
PreliminariesSome other topics:
Geometric graph theory
For example: Planarity, crossing number, ...
Topological combinatoricsFor example: Fair division problems, equipartitions
Valculescu Claudiu Discrete geometry from an algebraic point of view 4 / 25
PreliminariesSome other topics:
Geometric graph theoryFor example: Planarity, crossing number, ...
Topological combinatoricsFor example: Fair division problems, equipartitions
Valculescu Claudiu Discrete geometry from an algebraic point of view 4 / 25
PreliminariesSome other topics:
Geometric graph theoryFor example: Planarity, crossing number, ...
Topological combinatorics
For example: Fair division problems, equipartitions
Valculescu Claudiu Discrete geometry from an algebraic point of view 4 / 25
PreliminariesSome other topics:
Geometric graph theoryFor example: Planarity, crossing number, ...
Topological combinatoricsFor example: Fair division problems, equipartitions
Valculescu Claudiu Discrete geometry from an algebraic point of view 4 / 25
Preliminaries
Valculescu Claudiu Discrete geometry from an algebraic point of view 5 / 25
Preliminaries
Some other domains in which discrete geometry has applications:
Theoretical computer science
EngineeringFor example: Robotics;
Computer Imagery and Image processingFor example: Modeling image aggregation processes;
ChemistryFor example: Cristalography;
Other fields of mathematicsFor example: Numerical analysis.
Valculescu Claudiu Discrete geometry from an algebraic point of view 6 / 25
Preliminaries
Some other domains in which discrete geometry has applications:
Theoretical computer science
EngineeringFor example: Robotics;
Computer Imagery and Image processingFor example: Modeling image aggregation processes;
ChemistryFor example: Cristalography;
Other fields of mathematicsFor example: Numerical analysis.
Valculescu Claudiu Discrete geometry from an algebraic point of view 6 / 25
Preliminaries
Some other domains in which discrete geometry has applications:
Theoretical computer science
EngineeringFor example: Robotics;
Computer Imagery and Image processingFor example: Modeling image aggregation processes;
ChemistryFor example: Cristalography;
Other fields of mathematicsFor example: Numerical analysis.
Valculescu Claudiu Discrete geometry from an algebraic point of view 6 / 25
Preliminaries
Some other domains in which discrete geometry has applications:
Theoretical computer science
EngineeringFor example: Robotics;
Computer Imagery and Image processingFor example: Modeling image aggregation processes;
ChemistryFor example: Cristalography;
Other fields of mathematicsFor example: Numerical analysis.
Valculescu Claudiu Discrete geometry from an algebraic point of view 6 / 25
Preliminaries
Some other domains in which discrete geometry has applications:
Theoretical computer science
EngineeringFor example: Robotics;
Computer Imagery and Image processingFor example: Modeling image aggregation processes;
ChemistryFor example: Cristalography;
Other fields of mathematicsFor example: Numerical analysis.
Valculescu Claudiu Discrete geometry from an algebraic point of view 6 / 25
Preliminaries
Some other domains in which discrete geometry has applications:
Theoretical computer science
EngineeringFor example: Robotics;
Computer Imagery and Image processingFor example: Modeling image aggregation processes;
ChemistryFor example: Cristalography;
Other fields of mathematicsFor example: Numerical analysis.
Valculescu Claudiu Discrete geometry from an algebraic point of view 6 / 25
Big O and Ω notations
Definition (Big O notation)Let f and g be two functions defined on some subset of the real numbers.One writes
f (x) = O(g(x)),
if and only if there is a positive constant M and x0 > 0, such that
|f (x)| ≤ M|g(x)| for all x ≥ x0.
Definition (Ω notation)Let f and g be two functions defined on some subset of the real numbers.One writes
f (x) = Ω(g(x)),
if and only if there is a positive constant M and x0 > 0, such that
|f (x)| ≥ M|g(x)| for all x ≥ x0.
Valculescu Claudiu Discrete geometry from an algebraic point of view 7 / 25
Big O and Ω notations
Definition (Big O notation)Let f and g be two functions defined on some subset of the real numbers.One writes
f (x) = O(g(x)),
if and only if there is a positive constant M and x0 > 0, such that
|f (x)| ≤ M|g(x)| for all x ≥ x0.
Definition (Ω notation)Let f and g be two functions defined on some subset of the real numbers.One writes
f (x) = Ω(g(x)),
if and only if there is a positive constant M and x0 > 0, such that
|f (x)| ≥ M|g(x)| for all x ≥ x0.
Valculescu Claudiu Discrete geometry from an algebraic point of view 7 / 25
Big O and Ω notations
Definition (Big O notation)Let f and g be two functions defined on some subset of the real numbers.One writes
f (x) = O(g(x)),
if and only if there is a positive constant M and x0 > 0, such that
|f (x)| ≤ M|g(x)| for all x ≥ x0.
Definition (Ω notation)Let f and g be two functions defined on some subset of the real numbers.One writes
f (x) = Ω(g(x)),
if and only if there is a positive constant M and x0 > 0, such that
|f (x)| ≥ M|g(x)| for all x ≥ x0.
Valculescu Claudiu Discrete geometry from an algebraic point of view 7 / 25
Point-line incidencesRemember the point-line incidence question...
Theorem (Szemeredi-Trotter, 1983)Given a set P of n points and a set L of m lines in the real plane, thenumber of incidences is of order
O(n2/3m2/3 + m + n).
EndreSzemeredi
WilliamTrotter
Valculescu Claudiu Discrete geometry from an algebraic point of view 8 / 25
Point-line incidencesRemember the point-line incidence question...
Theorem (Szemeredi-Trotter, 1983)Given a set P of n points and a set L of m lines in the real plane, thenumber of incidences is of order
O(n2/3m2/3 + m + n).
EndreSzemeredi
WilliamTrotter
Valculescu Claudiu Discrete geometry from an algebraic point of view 8 / 25
Application of Szemeredi-Trotter
DefinitionGiven a finite set A ⊂ R, we define :
A + A = a + a′ : a, a′ ∈ A
A · A = a · a′ : a, a′ ∈ A.
Note the following!If A is an arithmetic progression a + bk, k = 1, ..., n, we have
|A + A| = O(|A|).
If A is a geometric progression a · bk , k = 1, ..., n, we have
|A · A| = O(|A|).
How ”small” can max|A + A|, |A · A| be?
Valculescu Claudiu Discrete geometry from an algebraic point of view 9 / 25
Application of Szemeredi-Trotter
DefinitionGiven a finite set A ⊂ R, we define :
A + A = a + a′ : a, a′ ∈ A
A · A = a · a′ : a, a′ ∈ A.
Note the following!
If A is an arithmetic progression a + bk, k = 1, ..., n, we have
|A + A| = O(|A|).
If A is a geometric progression a · bk , k = 1, ..., n, we have
|A · A| = O(|A|).
How ”small” can max|A + A|, |A · A| be?
Valculescu Claudiu Discrete geometry from an algebraic point of view 9 / 25
Application of Szemeredi-Trotter
DefinitionGiven a finite set A ⊂ R, we define :
A + A = a + a′ : a, a′ ∈ A
A · A = a · a′ : a, a′ ∈ A.
Note the following!If A is an arithmetic progression a + bk, k = 1, ..., n, we have
|A + A| = O(|A|).
If A is a geometric progression a · bk , k = 1, ..., n, we have
|A · A| = O(|A|).
How ”small” can max|A + A|, |A · A| be?
Valculescu Claudiu Discrete geometry from an algebraic point of view 9 / 25
Application of Szemeredi-Trotter
DefinitionGiven a finite set A ⊂ R, we define :
A + A = a + a′ : a, a′ ∈ A
A · A = a · a′ : a, a′ ∈ A.
Note the following!If A is an arithmetic progression a + bk, k = 1, ..., n, we have
|A + A| = O(|A|).
If A is a geometric progression a · bk , k = 1, ..., n, we have
|A · A| = O(|A|).
How ”small” can max|A + A|, |A · A| be?
Valculescu Claudiu Discrete geometry from an algebraic point of view 9 / 25
Application of Szemeredi-Trotter
DefinitionGiven a finite set A ⊂ R, we define :
A + A = a + a′ : a, a′ ∈ A
A · A = a · a′ : a, a′ ∈ A.
Note the following!If A is an arithmetic progression a + bk, k = 1, ..., n, we have
|A + A| = O(|A|).
If A is a geometric progression a · bk , k = 1, ..., n, we have
|A · A| = O(|A|).
How ”small” can max|A + A|, |A · A| be?Valculescu Claudiu Discrete geometry from an algebraic point of view 9 / 25
Application of Szemeredi-Trotter
Gyorgy Elekes
Theorem (Sum-Product, Elekes 1997)Let A ⊂ R be a finite set. Then
max|A + A|, |A · A| = Ω(|A|5/4).
Proof.Let P = (A · A)× (A + A).Let L be the set of all lines of the form y = ax + b, with a ∈ A−1 andb ∈ A (we can discard a = 0),where A−1 = x−1, x ∈ A.We have |L| = |A|2.Each line in L has at least |A| incidences with P.
Valculescu Claudiu Discrete geometry from an algebraic point of view 10 / 25
Application of Szemeredi-Trotter
Gyorgy Elekes
Theorem (Sum-Product, Elekes 1997)Let A ⊂ R be a finite set. Then
max|A + A|, |A · A| = Ω(|A|5/4).
Proof.
Let P = (A · A)× (A + A).Let L be the set of all lines of the form y = ax + b, with a ∈ A−1 andb ∈ A (we can discard a = 0),where A−1 = x−1, x ∈ A.We have |L| = |A|2.Each line in L has at least |A| incidences with P.
Valculescu Claudiu Discrete geometry from an algebraic point of view 10 / 25
Application of Szemeredi-Trotter
Gyorgy Elekes
Theorem (Sum-Product, Elekes 1997)Let A ⊂ R be a finite set. Then
max|A + A|, |A · A| = Ω(|A|5/4).
Proof.Let P = (A · A)× (A + A).
Let L be the set of all lines of the form y = ax + b, with a ∈ A−1 andb ∈ A (we can discard a = 0),where A−1 = x−1, x ∈ A.We have |L| = |A|2.Each line in L has at least |A| incidences with P.
Valculescu Claudiu Discrete geometry from an algebraic point of view 10 / 25
Application of Szemeredi-Trotter
Gyorgy Elekes
Theorem (Sum-Product, Elekes 1997)Let A ⊂ R be a finite set. Then
max|A + A|, |A · A| = Ω(|A|5/4).
Proof.Let P = (A · A)× (A + A).Let L be the set of all lines of the form y = ax + b, with a ∈ A−1 andb ∈ A (we can discard a = 0),
where A−1 = x−1, x ∈ A.We have |L| = |A|2.Each line in L has at least |A| incidences with P.
Valculescu Claudiu Discrete geometry from an algebraic point of view 10 / 25
Application of Szemeredi-Trotter
Gyorgy Elekes
Theorem (Sum-Product, Elekes 1997)Let A ⊂ R be a finite set. Then
max|A + A|, |A · A| = Ω(|A|5/4).
Proof.Let P = (A · A)× (A + A).Let L be the set of all lines of the form y = ax + b, with a ∈ A−1 andb ∈ A (we can discard a = 0),where A−1 = x−1, x ∈ A.
We have |L| = |A|2.Each line in L has at least |A| incidences with P.
Valculescu Claudiu Discrete geometry from an algebraic point of view 10 / 25
Application of Szemeredi-Trotter
Gyorgy Elekes
Theorem (Sum-Product, Elekes 1997)Let A ⊂ R be a finite set. Then
max|A + A|, |A · A| = Ω(|A|5/4).
Proof.Let P = (A · A)× (A + A).Let L be the set of all lines of the form y = ax + b, with a ∈ A−1 andb ∈ A (we can discard a = 0),where A−1 = x−1, x ∈ A.We have |L| = |A|2.
Each line in L has at least |A| incidences with P.
Valculescu Claudiu Discrete geometry from an algebraic point of view 10 / 25
Application of Szemeredi-Trotter
Gyorgy Elekes
Theorem (Sum-Product, Elekes 1997)Let A ⊂ R be a finite set. Then
max|A + A|, |A · A| = Ω(|A|5/4).
Proof.Let P = (A · A)× (A + A).Let L be the set of all lines of the form y = ax + b, with a ∈ A−1 andb ∈ A (we can discard a = 0),where A−1 = x−1, x ∈ A.We have |L| = |A|2.Each line in L has at least |A| incidences with P.
Valculescu Claudiu Discrete geometry from an algebraic point of view 10 / 25
Application of Szemeredi-Trotter
Thus, we have
|A · A|2/3 · |A + A|2/3 · |A|4/3 = Ω(|A|3).
Assume neither |A + A|, nor |A · A| are of order Ω(|A|5/4).
This leads us to a contradiction, and thus
max|A + A|, |A · A| = Ω(|A|5/4),
which completes the proof.
The conjectured bound is ∼ |A|2−ε, for all ε > 0.
Valculescu Claudiu Discrete geometry from an algebraic point of view 11 / 25
Application of Szemeredi-Trotter
Thus, we have
|A · A|2/3 · |A + A|2/3 · |A|4/3 = Ω(|A|3).
Assume neither |A + A|, nor |A · A| are of order Ω(|A|5/4).
This leads us to a contradiction, and thus
max|A + A|, |A · A| = Ω(|A|5/4),
which completes the proof.
The conjectured bound is ∼ |A|2−ε, for all ε > 0.
Valculescu Claudiu Discrete geometry from an algebraic point of view 11 / 25
Application of Szemeredi-Trotter
Thus, we have
|A · A|2/3 · |A + A|2/3 · |A|4/3 = Ω(|A|3).
Assume neither |A + A|, nor |A · A| are of order Ω(|A|5/4).
This leads us to a contradiction, and thus
max|A + A|, |A · A| = Ω(|A|5/4),
which completes the proof.
The conjectured bound is ∼ |A|2−ε, for all ε > 0.
Valculescu Claudiu Discrete geometry from an algebraic point of view 11 / 25
Application of Szemeredi-Trotter
Thus, we have
|A · A|2/3 · |A + A|2/3 · |A|4/3 = Ω(|A|3).
Assume neither |A + A|, nor |A · A| are of order Ω(|A|5/4).
This leads us to a contradiction, and thus
max|A + A|, |A · A| = Ω(|A|5/4),
which completes the proof.
The conjectured bound is ∼ |A|2−ε, for all ε > 0.
Valculescu Claudiu Discrete geometry from an algebraic point of view 11 / 25
Stepping forward...
Definition (Real algebraic curve)We call C ⊂ R2 an real algebraic curve if it isinfinite and there exist a polynomial f ∈ R[x , y ]\0such that
C = (x , y) ∈ R2 : f (x , y) = 0.
Lines, circles, ellipses, hyperbolas, parabolas are algebraic curves.
Definition (Degree of freedom. Multiplicity)Let P ⊆ RD be a set of points and let Γ be a set of curves in RD. We saythat P and Γ form a system with k degrees of freedom and multiplicity Mif any two curves in Γ intersect in at most M points of P, and any kpoints of P belong to at most M curves in Γ.
Valculescu Claudiu Discrete geometry from an algebraic point of view 12 / 25
Stepping forward...
Definition (Real algebraic curve)We call C ⊂ R2 an real algebraic curve if it isinfinite and there exist a polynomial f ∈ R[x , y ]\0such that
C = (x , y) ∈ R2 : f (x , y) = 0.
Lines, circles, ellipses, hyperbolas, parabolas are algebraic curves.
Definition (Degree of freedom. Multiplicity)Let P ⊆ RD be a set of points and let Γ be a set of curves in RD. We saythat P and Γ form a system with k degrees of freedom and multiplicity Mif any two curves in Γ intersect in at most M points of P, and any kpoints of P belong to at most M curves in Γ.
Valculescu Claudiu Discrete geometry from an algebraic point of view 12 / 25
Stepping forward...
Definition (Real algebraic curve)We call C ⊂ R2 an real algebraic curve if it isinfinite and there exist a polynomial f ∈ R[x , y ]\0such that
C = (x , y) ∈ R2 : f (x , y) = 0.
Lines, circles, ellipses, hyperbolas, parabolas are algebraic curves.
Definition (Degree of freedom. Multiplicity)Let P ⊆ RD be a set of points and let Γ be a set of curves in RD. We saythat P and Γ form a system with k degrees of freedom and multiplicity Mif any two curves in Γ intersect in at most M points of P, and any kpoints of P belong to at most M curves in Γ.
Valculescu Claudiu Discrete geometry from an algebraic point of view 12 / 25
Stepping forward...
Theorem (Pach-Sharir, 1998)If a set P of points in R2 and a set Γ of algebraic curves in R2 form asystem with 2 degrees of freedom and multiplicity M, then
I(P, Γ) ≤ CM ·max|P|2/3|Γ|2/3, |P|, |Γ|,
where CM is a constant depending only on M.
Janos Pach Micha SharirValculescu Claudiu Discrete geometry from an algebraic point of view 13 / 25
Applications of algebraic geometry
Paul Erdos
Problem (Erdos, 1946)Prove that any set of n points in the planedetermines at least n1−ε distinct distances.
Solved - 2010 - Guth, Katz - using tools from algebraic geometry.
Larry Guth Nets Hawk Katz
Valculescu Claudiu Discrete geometry from an algebraic point of view 14 / 25
Applications of algebraic geometry
Paul Erdos
Problem (Erdos, 1946)Prove that any set of n points in the planedetermines at least n1−ε distinct distances.
Solved - 2010 - Guth, Katz - using tools from algebraic geometry.
Larry Guth Nets Hawk KatzValculescu Claudiu Discrete geometry from an algebraic point of view 14 / 25
Variation of Erdos’ distinct distance problem
Theorem (Pach-De Zeeuw, 2013)Given a plane algebraic curve C of degree d with n points on it, thenumber of distinct distances spanned by the set of points is Ωd (n4/3),unless C contains a line or a circle.
Janos Pach Frank de Zeeuw
Remark: If the curve contains a line or a circle, then there areconstructions for which the number of distances spanned is linear!
Valculescu Claudiu Discrete geometry from an algebraic point of view 15 / 25
Variation of Erdos’ distinct distance problem
Theorem (Pach-De Zeeuw, 2013)Given a plane algebraic curve C of degree d with n points on it, thenumber of distinct distances spanned by the set of points is Ωd (n4/3),unless C contains a line or a circle.
Janos Pach Frank de Zeeuw
Remark: If the curve contains a line or a circle, then there areconstructions for which the number of distances spanned is linear!
Valculescu Claudiu Discrete geometry from an algebraic point of view 15 / 25
Distinct values of bilinear functions on algebraic curves
Definition (Complex algebraic curve)We call C ⊂ C2 a complex algebraic curve if there exist a polynomialf ∈ C[x , y ]\0 such that
C = (x , y) ∈ C2 : f (x , y) = 0.
For all 2× 2 matrices M with complex entries, let BM : C2 × C2 → C bedefined as
BM(p, q) = pT Mq,∀p, q ∈ C2.
DefinitionWe call a curve in C2 special if it is a line or it is linearly equivalent to acurve defined by an equation of the form
xk = y l , k, l ∈ Z, (k, l) = 1.
Valculescu Claudiu Discrete geometry from an algebraic point of view 16 / 25
Distinct values of bilinear functions on algebraic curves
Definition (Complex algebraic curve)We call C ⊂ C2 a complex algebraic curve if there exist a polynomialf ∈ C[x , y ]\0 such that
C = (x , y) ∈ C2 : f (x , y) = 0.
For all 2× 2 matrices M with complex entries, let BM : C2 × C2 → C bedefined as
BM(p, q) = pT Mq,∀p, q ∈ C2.
DefinitionWe call a curve in C2 special if it is a line or it is linearly equivalent to acurve defined by an equation of the form
xk = y l , k, l ∈ Z, (k, l) = 1.
Valculescu Claudiu Discrete geometry from an algebraic point of view 16 / 25
Distinct values of bilinear functions on algebraic curves
Definition (Complex algebraic curve)We call C ⊂ C2 a complex algebraic curve if there exist a polynomialf ∈ C[x , y ]\0 such that
C = (x , y) ∈ C2 : f (x , y) = 0.
For all 2× 2 matrices M with complex entries, let BM : C2 × C2 → C bedefined as
BM(p, q) = pT Mq,∀p, q ∈ C2.
DefinitionWe call a curve in C2 special if it is a line or it is linearly equivalent to acurve defined by an equation of the form
xk = y l , k, l ∈ Z, (k, l) = 1.
Valculescu Claudiu Discrete geometry from an algebraic point of view 16 / 25
Distinct values of bilinear functions on algebraic curves
Definition (Complex algebraic curve)We call C ⊂ C2 a complex algebraic curve if there exist a polynomialf ∈ C[x , y ]\0 such that
C = (x , y) ∈ C2 : f (x , y) = 0.
For all 2× 2 matrices M with complex entries, let BM : C2 × C2 → C bedefined as
BM(p, q) = pT Mq,∀p, q ∈ C2.
DefinitionWe call a curve in C2 special if it is a line or it is linearly equivalent to acurve defined by an equation of the form
xk = y l , k, l ∈ Z, (k, l) = 1.
Valculescu Claudiu Discrete geometry from an algebraic point of view 16 / 25
Distinct values of bilinear functions on algebraic curves
Theorem (Valculescu-De Zeeuw, 2014)Let C be an irreducible algebraic curve in C of degree d, S a finite set ofpoints on C. If BM is defined as above, then the following holds:
|BM(S)| = Ω(d−5|S|4/3),
where BM(S) = B(p, q) : p, q ∈ S, unless C is special or M is singular.
Remark. If M is singular or C is special, there are constructions where thenumber of distinct values is linear in |S|.
Sketch of proof:Idea: reduce the problem to an incidence problem (introduced by Elekes).The same idea was used by Guth and Katz in their solution for the distinctdistance problem.
Valculescu Claudiu Discrete geometry from an algebraic point of view 17 / 25
Distinct values of bilinear functions on algebraic curves
Theorem (Valculescu-De Zeeuw, 2014)Let C be an irreducible algebraic curve in C of degree d, S a finite set ofpoints on C. If BM is defined as above, then the following holds:
|BM(S)| = Ω(d−5|S|4/3),
where BM(S) = B(p, q) : p, q ∈ S, unless C is special or M is singular.
Remark. If M is singular or C is special, there are constructions where thenumber of distinct values is linear in |S|.
Sketch of proof:Idea: reduce the problem to an incidence problem (introduced by Elekes).The same idea was used by Guth and Katz in their solution for the distinctdistance problem.
Valculescu Claudiu Discrete geometry from an algebraic point of view 17 / 25
Distinct values of bilinear functions on algebraic curves
Theorem (Valculescu-De Zeeuw, 2014)Let C be an irreducible algebraic curve in C of degree d, S a finite set ofpoints on C. If BM is defined as above, then the following holds:
|BM(S)| = Ω(d−5|S|4/3),
where BM(S) = B(p, q) : p, q ∈ S, unless C is special or M is singular.
Remark. If M is singular or C is special, there are constructions where thenumber of distinct values is linear in |S|.
Sketch of proof:
Idea: reduce the problem to an incidence problem (introduced by Elekes).The same idea was used by Guth and Katz in their solution for the distinctdistance problem.
Valculescu Claudiu Discrete geometry from an algebraic point of view 17 / 25
Distinct values of bilinear functions on algebraic curves
Theorem (Valculescu-De Zeeuw, 2014)Let C be an irreducible algebraic curve in C of degree d, S a finite set ofpoints on C. If BM is defined as above, then the following holds:
|BM(S)| = Ω(d−5|S|4/3),
where BM(S) = B(p, q) : p, q ∈ S, unless C is special or M is singular.
Remark. If M is singular or C is special, there are constructions where thenumber of distinct values is linear in |S|.
Sketch of proof:Idea: reduce the problem to an incidence problem (introduced by Elekes).
The same idea was used by Guth and Katz in their solution for the distinctdistance problem.
Valculescu Claudiu Discrete geometry from an algebraic point of view 17 / 25
Distinct values of bilinear functions on algebraic curves
Theorem (Valculescu-De Zeeuw, 2014)Let C be an irreducible algebraic curve in C of degree d, S a finite set ofpoints on C. If BM is defined as above, then the following holds:
|BM(S)| = Ω(d−5|S|4/3),
where BM(S) = B(p, q) : p, q ∈ S, unless C is special or M is singular.
Remark. If M is singular or C is special, there are constructions where thenumber of distinct values is linear in |S|.
Sketch of proof:Idea: reduce the problem to an incidence problem (introduced by Elekes).The same idea was used by Guth and Katz in their solution for the distinctdistance problem.
Valculescu Claudiu Discrete geometry from an algebraic point of view 17 / 25
Distinct values of bilinear functions on algebraic curves
Let BM(S) = B(pi , qj) : pi , qj ∈ S.
Let Q = (p, p′, q, q′) ∈ S4 : BM(p, q) = BM(p′, q′).
For each value a ∈ BM(S), let
Ea = (pi , qs) ∈ S2 : BM(pi , qs) = a.
Using Cauchy-Schwarz, we have
|Q| =∑
a∈BM(S)|Ea|2 ≥
1BM(S)
∑a∈BM(S)
|Ea|
2
= n4
BM(S) .
Valculescu Claudiu Discrete geometry from an algebraic point of view 18 / 25
Distinct values of bilinear functions on algebraic curves
Let BM(S) = B(pi , qj) : pi , qj ∈ S.
Let Q = (p, p′, q, q′) ∈ S4 : BM(p, q) = BM(p′, q′).
For each value a ∈ BM(S), let
Ea = (pi , qs) ∈ S2 : BM(pi , qs) = a.
Using Cauchy-Schwarz, we have
|Q| =∑
a∈BM(S)|Ea|2 ≥
1BM(S)
∑a∈BM(S)
|Ea|
2
= n4
BM(S) .
Valculescu Claudiu Discrete geometry from an algebraic point of view 18 / 25
Distinct values of bilinear functions on algebraic curves
Let BM(S) = B(pi , qj) : pi , qj ∈ S.
Let Q = (p, p′, q, q′) ∈ S4 : BM(p, q) = BM(p′, q′).
For each value a ∈ BM(S), let
Ea = (pi , qs) ∈ S2 : BM(pi , qs) = a.
Using Cauchy-Schwarz, we have
|Q| =∑
a∈BM(S)|Ea|2 ≥
1BM(S)
∑a∈BM(S)
|Ea|
2
= n4
BM(S) .
Valculescu Claudiu Discrete geometry from an algebraic point of view 18 / 25
Distinct values of bilinear functions on algebraic curves
Let BM(S) = B(pi , qj) : pi , qj ∈ S.
Let Q = (p, p′, q, q′) ∈ S4 : BM(p, q) = BM(p′, q′).
For each value a ∈ BM(S), let
Ea = (pi , qs) ∈ S2 : BM(pi , qs) = a.
Using Cauchy-Schwarz, we have
|Q| =∑
a∈BM(S)|Ea|2 ≥
1BM(S)
∑a∈BM(S)
|Ea|
2
= n4
BM(S) .
Valculescu Claudiu Discrete geometry from an algebraic point of view 18 / 25
Distinct values of bilinear functions on algebraic curves
Define the following curves:Cij = (q, q′) ∈ C2 : BM(pi , q) = BM(pj , q′).
Cst = (p, p′) ∈ C2 : BM(p, qs) = BM(p′, qt).Let Γ be the set of all the curves Cij .A point (qs , qt) ∈ P lies on Cij iff (pi , pj , qs , qt) ∈ Q.That means
n4
|BM(S)| ≤ |Q| = I(P, Γ) ≤?.
An upper bound for|I(P, Γ)| will give us a lower
bound on |BM(S)|.
Valculescu Claudiu Discrete geometry from an algebraic point of view 19 / 25
Distinct values of bilinear functions on algebraic curvesDefine the following curves:
Cij = (q, q′) ∈ C2 : BM(pi , q) = BM(pj , q′).
Cst = (p, p′) ∈ C2 : BM(p, qs) = BM(p′, qt).
Let Γ be the set of all the curves Cij .A point (qs , qt) ∈ P lies on Cij iff (pi , pj , qs , qt) ∈ Q.That means
n4
|BM(S)| ≤ |Q| = I(P, Γ) ≤?.
An upper bound for|I(P, Γ)| will give us a lower
bound on |BM(S)|.
Valculescu Claudiu Discrete geometry from an algebraic point of view 19 / 25
Distinct values of bilinear functions on algebraic curvesDefine the following curves:
Cij = (q, q′) ∈ C2 : BM(pi , q) = BM(pj , q′).
Cst = (p, p′) ∈ C2 : BM(p, qs) = BM(p′, qt).Let Γ be the set of all the curves Cij .
A point (qs , qt) ∈ P lies on Cij iff (pi , pj , qs , qt) ∈ Q.That means
n4
|BM(S)| ≤ |Q| = I(P, Γ) ≤?.
An upper bound for|I(P, Γ)| will give us a lower
bound on |BM(S)|.
Valculescu Claudiu Discrete geometry from an algebraic point of view 19 / 25
Distinct values of bilinear functions on algebraic curvesDefine the following curves:
Cij = (q, q′) ∈ C2 : BM(pi , q) = BM(pj , q′).
Cst = (p, p′) ∈ C2 : BM(p, qs) = BM(p′, qt).Let Γ be the set of all the curves Cij .A point (qs , qt) ∈ P lies on Cij iff (pi , pj , qs , qt) ∈ Q.
That means
n4
|BM(S)| ≤ |Q| = I(P, Γ) ≤?.
An upper bound for|I(P, Γ)| will give us a lower
bound on |BM(S)|.
Valculescu Claudiu Discrete geometry from an algebraic point of view 19 / 25
Distinct values of bilinear functions on algebraic curvesDefine the following curves:
Cij = (q, q′) ∈ C2 : BM(pi , q) = BM(pj , q′).
Cst = (p, p′) ∈ C2 : BM(p, qs) = BM(p′, qt).Let Γ be the set of all the curves Cij .A point (qs , qt) ∈ P lies on Cij iff (pi , pj , qs , qt) ∈ Q.That means
n4
|BM(S)| ≤ |Q| = I(P, Γ)
≤?.
An upper bound for|I(P, Γ)| will give us a lower
bound on |BM(S)|.
Valculescu Claudiu Discrete geometry from an algebraic point of view 19 / 25
Distinct values of bilinear functions on algebraic curvesDefine the following curves:
Cij = (q, q′) ∈ C2 : BM(pi , q) = BM(pj , q′).
Cst = (p, p′) ∈ C2 : BM(p, qs) = BM(p′, qt).Let Γ be the set of all the curves Cij .A point (qs , qt) ∈ P lies on Cij iff (pi , pj , qs , qt) ∈ Q.That means
n4
|BM(S)| ≤ |Q| = I(P, Γ) ≤?.
An upper bound for|I(P, Γ)| will give us a lower
bound on |BM(S)|.Valculescu Claudiu Discrete geometry from an algebraic point of view 19 / 25
Distinct values of bilinear functions on algebraic curves
The desired bound is given by the following generalization of thePach-Sharir incidence theorem.
Theorem (Solymosi-De Zeeuw, 2014)Let A,B ⊂ C2 with |A| = |B|, let P = A× B ⊂ C4, and let Γ be a set ofalgebraic curves of degree at most d2 in C4, with |P| = |Γ| = m2. If anytwo points of P are contained in at most d2 curves of Γ, then we have
I(P, Γ) = O(d10/3m8/3).
The points and curves defined above might not fulfill the conditions of thetheorem...
We have to remove some curves and some points!
Valculescu Claudiu Discrete geometry from an algebraic point of view 20 / 25
Distinct values of bilinear functions on algebraic curves
The desired bound is given by the following generalization of thePach-Sharir incidence theorem.
Theorem (Solymosi-De Zeeuw, 2014)Let A,B ⊂ C2 with |A| = |B|, let P = A× B ⊂ C4, and let Γ be a set ofalgebraic curves of degree at most d2 in C4, with |P| = |Γ| = m2. If anytwo points of P are contained in at most d2 curves of Γ, then we have
I(P, Γ) = O(d10/3m8/3).
The points and curves defined above might not fulfill the conditions of thetheorem...
We have to remove some curves and some points!
Valculescu Claudiu Discrete geometry from an algebraic point of view 20 / 25
Distinct values of bilinear functions on algebraic curves
The desired bound is given by the following generalization of thePach-Sharir incidence theorem.
Theorem (Solymosi-De Zeeuw, 2014)Let A,B ⊂ C2 with |A| = |B|, let P = A× B ⊂ C4, and let Γ be a set ofalgebraic curves of degree at most d2 in C4, with |P| = |Γ| = m2. If anytwo points of P are contained in at most d2 curves of Γ, then we have
I(P, Γ) = O(d10/3m8/3).
The points and curves defined above might not fulfill the conditions of thetheorem...
We have to remove some curves and some points!
Valculescu Claudiu Discrete geometry from an algebraic point of view 20 / 25
Distinct values of bilinear functions on algebraic curves
The desired bound is given by the following generalization of thePach-Sharir incidence theorem.
Theorem (Solymosi-De Zeeuw, 2014)Let A,B ⊂ C2 with |A| = |B|, let P = A× B ⊂ C4, and let Γ be a set ofalgebraic curves of degree at most d2 in C4, with |P| = |Γ| = m2. If anytwo points of P are contained in at most d2 curves of Γ, then we have
I(P, Γ) = O(d10/3m8/3).
The points and curves defined above might not fulfill the conditions of thetheorem...
We have to remove some curves and some points!
Valculescu Claudiu Discrete geometry from an algebraic point of view 20 / 25
Distinct values of bilinear functions on algebraic curves
What points and curves should we remove?
No, we will remove only the following:
Γ0 = Cij ∈ Γ : ∃Ckl ∈ Γ : |Cij ∩ Ckl | =∞,
P0 = (qs , qt) ∈ P : ∃(qu, qv ) ∈ P : |Cst ∩ Cuv | =∞.
Valculescu Claudiu Discrete geometry from an algebraic point of view 21 / 25
Distinct values of bilinear functions on algebraic curves
What points and curves should we remove?
No, we will remove only the following:
Γ0 = Cij ∈ Γ : ∃Ckl ∈ Γ : |Cij ∩ Ckl | =∞,
P0 = (qs , qt) ∈ P : ∃(qu, qv ) ∈ P : |Cst ∩ Cuv | =∞.
Valculescu Claudiu Discrete geometry from an algebraic point of view 21 / 25
Distinct values of bilinear functions on algebraic curves
We want to show that Γ0 and P0 are relatively small!
We do this by showing that if two curves have infinite intersection, thenthis is related to an automorphism of C (invertible linear transformationthat fixes the curve).
One can prove that an irreducible algebraic curve of degree d has atmost d7 linear automorphisms, unless it is a special curve, whichcompletes the proof.
What about special curves?
Valculescu Claudiu Discrete geometry from an algebraic point of view 22 / 25
Distinct values of bilinear functions on algebraic curves
We want to show that Γ0 and P0 are relatively small!
We do this by showing that if two curves have infinite intersection, thenthis is related to an automorphism of C (invertible linear transformationthat fixes the curve).
One can prove that an irreducible algebraic curve of degree d has atmost d7 linear automorphisms, unless it is a special curve, whichcompletes the proof.
What about special curves?
Valculescu Claudiu Discrete geometry from an algebraic point of view 22 / 25
Distinct values of bilinear functions on algebraic curves
We want to show that Γ0 and P0 are relatively small!
We do this by showing that if two curves have infinite intersection, thenthis is related to an automorphism of C (invertible linear transformationthat fixes the curve).
One can prove that an irreducible algebraic curve of degree d has atmost d7 linear automorphisms, unless it is a special curve, whichcompletes the proof.
What about special curves?
Valculescu Claudiu Discrete geometry from an algebraic point of view 22 / 25
Distinct values of bilinear functions on algebraic curves
We want to show that Γ0 and P0 are relatively small!
We do this by showing that if two curves have infinite intersection, thenthis is related to an automorphism of C (invertible linear transformationthat fixes the curve).
One can prove that an irreducible algebraic curve of degree d has atmost d7 linear automorphisms, unless it is a special curve, whichcompletes the proof.
What about special curves?
Valculescu Claudiu Discrete geometry from an algebraic point of view 22 / 25
Distinct values of bilinear functions on algebraic curves
In fact, yes. Just consider the inner product, given by
BI(p, q) = xpxq + ypyq.
On a special curve, the number of distinct values of BI can be linear in thesize of the set of points. Also, if M is singular, the number of distinctvalues can be linear.
Valculescu Claudiu Discrete geometry from an algebraic point of view 23 / 25
Distinct values of bilinear functions on algebraic curves
In fact, yes.
Just consider the inner product, given by
BI(p, q) = xpxq + ypyq.
On a special curve, the number of distinct values of BI can be linear in thesize of the set of points. Also, if M is singular, the number of distinctvalues can be linear.
Valculescu Claudiu Discrete geometry from an algebraic point of view 23 / 25
Distinct values of bilinear functions on algebraic curves
In fact, yes. Just consider the inner product, given by
BI(p, q) = xpxq + ypyq.
On a special curve, the number of distinct values of BI can be linear in thesize of the set of points. Also, if M is singular, the number of distinctvalues can be linear.
Valculescu Claudiu Discrete geometry from an algebraic point of view 23 / 25
Distinct values of bilinear functions on algebraic curves
In fact, yes. Just consider the inner product, given by
BI(p, q) = xpxq + ypyq.
On a special curve, the number of distinct values of BI can be linear in thesize of the set of points. Also, if M is singular, the number of distinctvalues can be linear.
Valculescu Claudiu Discrete geometry from an algebraic point of view 23 / 25
Distinct values of bilinear functions on algebraic curves
Examples of constructions spanning linear number of distinct values:
Let us defineS := (2li , 2ki ) : i = 1, ..., |S|.
ThenBI((2li , 2ki ), (2lj , 2kj)) = (2l )i+j + (2k)i+j .
That means
|BI(S)| = 2|S| − 1.
Thus, the number of distinct values is linear in |S|.
Valculescu Claudiu Discrete geometry from an algebraic point of view 24 / 25
Distinct values of bilinear functions on algebraic curves
Examples of constructions spanning linear number of distinct values:
Let us defineS := (2li , 2ki ) : i = 1, ..., |S|.
ThenBI((2li , 2ki ), (2lj , 2kj)) = (2l )i+j + (2k)i+j .
That means
|BI(S)| = 2|S| − 1.
Thus, the number of distinct values is linear in |S|.
Valculescu Claudiu Discrete geometry from an algebraic point of view 24 / 25
Distinct values of bilinear functions on algebraic curves
Examples of constructions spanning linear number of distinct values:
Let us defineS := (2li , 2ki ) : i = 1, ..., |S|.
ThenBI((2li , 2ki ), (2lj , 2kj)) = (2l )i+j + (2k)i+j .
That means
|BI(S)| = 2|S| − 1.
Thus, the number of distinct values is linear in |S|.
Valculescu Claudiu Discrete geometry from an algebraic point of view 24 / 25
Distinct values of bilinear functions on algebraic curves
Examples of constructions spanning linear number of distinct values:
Let us defineS := (2li , 2ki ) : i = 1, ..., |S|.
ThenBI((2li , 2ki ), (2lj , 2kj)) = (2l )i+j + (2k)i+j .
That means
|BI(S)| = 2|S| − 1.
Thus, the number of distinct values is linear in |S|.
Valculescu Claudiu Discrete geometry from an algebraic point of view 24 / 25
Distinct values of bilinear functions on algebraic curves
Examples of constructions spanning linear number of distinct values:
Let us defineS := (2li , 2ki ) : i = 1, ..., |S|.
ThenBI((2li , 2ki ), (2lj , 2kj)) = (2l )i+j + (2k)i+j .
That means
|BI(S)| = 2|S| − 1.
Thus, the number of distinct values is linear in |S|.
Valculescu Claudiu Discrete geometry from an algebraic point of view 24 / 25
Thank you!
Valculescu Claudiu Discrete geometry from an algebraic point of view 25 / 25