Ten Points on a Cubic
Will Traves
Department of Mathematics
United States Naval Academy
Commutative Algebra meets Algebraic Combinatorics
Queen’s University, Kingston
25 JAN 2015
Report on joint work with David Wehlau
Traves (USNA)
CAAC 2015
25 JAN 2015
Easy Warm Up
There exists a unique curve of degree d through
general position in the plane.
x0 y0 z0 x1 y1 z1 x y z [012] = 0
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2
x x0 y0 x0 z0 y 2 y0 z0 z 2 0
0
0
x 2 x1 y1 x1 z1 y 2 y1 z1 z 2 1
1
1
..
..
..
..
..
.. .
.
.
.
.
.
x 2 xy
xz y 2 yz z 2 [6x6, d = 2] = 0
CAAC 2015
d+2
2
− 1 points in
10x10
cubic monos in x, y, and z [10x10, d = 3] = 0
25 JAN 2015
Brackets
2
With n points, the 3x3 determinants [ijk] generate C[(P2 )n ]Aut(P ) .
The relations among the brackets are generated by Plücker relations,
quadratic relations in the [ijk].
Determinantal formula can be rewritten in terms of brackets:
[6x6, d = 2] = [012][045][315][342] − [345][312][042][015]
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Pascal’s Theorem and the Grassmann-Cayley Algebra
There is also a constructive algorithm, Pascal’s Theorem, to check if 6
points lie on a conic.
2
The Grassmann-Cayley algebra GC ⊂ C[(P2 )n ]Aut(P ) consists of
expressions that can be expressed in terms of synthetic projective
geometry (Rota: Meet and Join Algebra).
Pascal’s Theorem: (04 ∧ 13) ∨ (05 ∧ 23) ∨ (24 ∧ 15) = 0.
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Theorem (Sturmfels and Whiteley)
For each polynomial P in the brackets, there is a monomial m in the
brackets so that mP ∈ GC.
Corollary
Given any configuration of plane points so that no 3 lie on a line then
any invariant property of these points can be verified using a
straightedge construction (i.e. by drawing and intersecting lines).
Traves (USNA)
CAAC 2015
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10 points on a cubic
Question
Can we find explicit constructions that test whether 10 points lie on a
cubic?
Reiss (1842): Factored the 10x10 determinant formula into a 20 term
degree 10 polynomial in the brackets. A “tour de force” of the symbolic
method.
David Wehlau (and his Mac) checked this factorization in 2 ways:
Evaluated both expressions at 100 sets of 10 points (seconds)
Expanded both expressions (25 million terms; hours).
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CAAC 2015
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The Cayley-Bacharach Theorem
The Cayley-Bacharach Theorem is a vast generalization of the 8 ⇒ 9
theorem in algebraic geometry.
Applying the CBT to 10 points:
1. split 10 points into two groups of 5,
giving red and blue degree 4 curves
2. find intersections of red and blue
curves
3. 10 points on a cubic ⇐⇒
6 auxiliary points on a conic
4. Use Pascal’s Theorem to check the
conic condition with a ruler.
Traves (USNA)
CAAC 2015
25 JAN 2015
Application of Cayley-Bacharach Theorem
Two degree 4 curves meet in 16 points, the original 10 and 6 residual
points.
4+4-3-3 = 2
dim
vs of deg-3 through 10 pts
vs of deg-3 through 16 pts
= failure of 6 pts on deg-2 curves
left side ≥ 1 ⇐⇒ deg-3 through 10 pts
failure on right side ≥ 1 ⇐⇒ deg-2 through 6 pts
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conic(A, B, C, D, E) ∩ conic(A, B, C, D 0 , E 0 )
Define Cremona(A,B,C;P) = line through AP ∩ BC and CP ∩ AB.
Find PDE = Cremona(A,B,C;D) ∩ Cremona(A,B,C;E)
and PD 0 E 0 = Cremona(A,B,C;D’) ∩ Cremona(A,B,C;E’)
Line joining PDE to PD 0 E 0 is Cremona(A,B,C;N),
where N is the fourth point of intersection of the conics.
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CAAC 2015
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conic(A, B, C, D, E) ∩ conic(A, B, C 0 , D 0 , E 0 )
Every circle passes through I(i : 1 : 0) and J(−i : 1 : 0).
√
Can’t construct the two points of intersection with a ruler (since 2 3 is
not in Q(i)).
Can construct the intersection of two circles with ruler and compass.
The analogous algorithm to intersect two conics sharing two points can
be made explicit.
Traves (USNA)
CAAC 2015
25 JAN 2015
Constructive Results
Theorem (T-, Wehlau)
Given 10 points in the plane there is an explicit ruler and compass
construction to check if the 10 points lie on a cubic.
When 6 of the 10 points lie on a conic then can test whether the 10
points lie on a cubic using only a straightedge.
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The Lingering Question
Sturmfels and Whiteley’s result guarantees the existence of a
straightedge construction that checks whether 10 points lie on a cubic
(when no 3 points are collinear).
Question
Can this construction be made explicit using a reasonable number of
lines?
In fact, Sturmfels and Whiteley’s theorem is constructive but the
construction produced by their proof requires about 100 million lines!
Can we exploit the group law on the cubic to assist in the construction?
Traves (USNA)
CAAC 2015
25 JAN 2015
Extensions
Question
When do
d+2
2
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 d+2
points lie on a degree d curve ⇐⇒ the
2
points lie on a degree d − 1 curve.
d+1
2
residual
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
25 JAN 2015