Ten Points on a Cubic
Will Traves
Department of Mathematics
United States Naval Academy
MAA MathFest
Washington, D.C.
7 AUG 2015
Report on joint work with David Wehlau (RMC-C)
Traves (USNA)
CAAC 2015
7 AUG 2015
Conics interpolating points in the plane
There exists an ellipse, hyperbola or pair of crossed lines through any
5 points in the plane.
Theorem (Pascal’s Mystic Hexagon Theorem, 1640)
Given 6 points, there exists a conic through all 6
⇐⇒ 3 auxilliary points are collinear.
Traves (USNA)
CAAC 2015
7 AUG 2015
Why is the theorem “mystic”?
Traves (USNA)
CAAC 2015
7 AUG 2015
Cubics interpolating points in the plane
There is at least one degree 3 curve through every set of 9 points.
Question
When do 10 points lie on a plane curve of degree 3?
Smooth curves of degree 3 are called elliptic curves and play a role in
both elliptic curve cryptography and in Wiles’s proof of Fermat’s
Last Theorem.
Traves (USNA)
CAAC 2015
7 AUG 2015
The Key Idea: Cayley-Bacharach
10 points on a cubic precisely when 6 auxiliary points on a conic.
We construct the 6 points using straightedge and compass and then
invoke Pascal’s Theorem.
Traves (USNA)
CAAC 2015
7 AUG 2015
3⇒4
Question
If we know 3 of the four points of intersection of two conics, can we find
the other using only a straightedge?
Traves (USNA)
CAAC 2015
7 AUG 2015
2⇒4
Question
If we know 2 points of intersecton of two conics, can we find the other
two using a compass and straightedge?
Idea: move points of intersection to [i : 1 : 0] and [−i : 1 : 0], when the
conics become circles. Check that the construction still makes sense
without the conics in special position.
Note: Our straightedge and compass must be complex, e.g. to
intersect non-overlapping circles.
Traves (USNA)
CAAC 2015
7 AUG 2015
Results
Theorem (T- and Wehlau)
There is a straightedge and compass construction to check whether
10 points lie on a cubic curve.
Theorem (T- and Wehlau)
When 6 of the 10 points lie on a conic then there is a
straightedge-only construction to check whether all 10 points lie on a
cubic curve.
Question
Does there exist a straightedge-only construction under weaker
hypotheses?
Traves (USNA)
CAAC 2015
7 AUG 2015
Straightedge-only construction
Theorem (Sturmfels and Whiteley)
There exists a straightedge-only construction to determine if 10
points lie on a cubic.
Proof depends on Mnëv’s Universality Theorem and constructs the
required lines! The algorithm uses about 100 million lines.
Can we exploit the group law on the cubic to assist in the construction?
Traves (USNA)
CAAC 2015
7 AUG 2015
Extensions
Question
d+2
2
When do
points lie on a degree d curve?
Take two degree d + 1 curves through the points;
these intersect in (d + 1)2 points, leaving d+1
residual points.
2
Theorem
The
the
d+2
2
d+1
2
points lie on a degree d curve ⇐⇒
residual points lie on a degree d − 1 curve.
This sets up an induction but to make the result constructive you
need algorithms to intersect curves defined by points alone.
We’re currently trying to do this explicitly in the degree 4 case.
Traves (USNA)
CAAC 2015
7 AUG 2015