# Current Research projects

## D21 - Find all Heron triangles with three rational medians

* Prove all remaining Buchholz-Rathbun triangles have irrational 3rd median (D21)
... calculate mappings C_i --> C_4, i=1..8
... calculate bad primes of genus 7 variety C_4
... calculate non-singular model of C_4
... calculate rank of jacobian of C_4
... calculate all rational points of C_4

* Speed up the search code for Heron-2-median triangles
... 2 median parametrization
... 1 median parametrization & test area, g(a,b,c) = (p^2+2q^2+2r^2-4pr,2q(p-2r),-p^2+2q^2-2r^2+2rp)
... Heron parametrization & test 1 median g(a,b,c) = (n(m^2+k^2),m(n^2+k^2),(n+m)(mn-k^2))

* Study Elkies procedure & apply to sporadic H2M triangles
... (x,y) = (p(z),p'(z)),j=j(t) ==> y^2 = sp^2, ==> SWNF

* Build new (theta,phi) points from smooth integers
* Characterize heights of (theta,phi) points
* Are the Eulerian parametrization triangles ever Heron? (triangle), (triangle.out)
* Are the Kraitchik parametrization triangles ever Heron?
* Are the Cole parametrization triangles ever Heron?
* Compare the 37 point cubic to the H2M elliptic curve
* Plot H2M triangles in barycentric coordinates
* Writeup the Area=0(mod 7) proof

## Find all cyclic polygons with rational area

* n=4 : Find all Brahmagupta quadrilaterals with Area=(Perimeter/4)^2
* n=5 : Are all Robbins' pentagons decomposable?
... Find all Robbins' pentagons with Area=(Perimeter/5)^2
... Find cyclic 5-gons with 5 rational medians
... prove 6 divides the area
* n=6 : Do there exist diagonally indecomposable Robbins' hexagons?
... Find all Robbins' hexagons with Area=(Perimeter/6)^2
* n=7 : Find the smallest Robbins' heptagon
... Do there exist non-Eulerian Robbins' heptagon
... Consider area formula modulo 2 ==> perimeter constraint
* n : Prove Radial and Quadrilateral decompositions are only two types

## Find all rational derived polynomials

* study nice quartics surface
* increase search space for p_(1,1,1,1) quartics
* reconsider degenerate vs. non-degenerate p_(1,1,1,1) quartics

## Elliptic curve theory

* complete the BALROG

## Triangles - Miscellaneous problems

* Writeup 2 vincent,2 median isosceles <==> C: y^2 = 54x^4-20x^2+2, (7/15,34/75)
* Do there exist 2-angle bisector,1-median other than (833,832,441)?
* Do there exist 1-angle bisectors,1-median,1-vincent other than (1099,640,529)?
* What is the rank of elliptic curves corresponding to Area=(Perimeter/k)^2, k=1,2,...?

## Papers to write or finish

* n-gon diagonal quartic paper with Bud
* Heron-2-median symmetries
* Pseudopowerful numbers
* No-3-line
* Spiral polygons (6-cycle.magma)
* n-simplex formula
* Square-like triangles
* Heron tetrahedra
* Infinity heresy
* Asymptotic Series : Harridge Problem

Ralph Heiner Buchholz
20-Feb-2008