Intersection of quadrics and Intersection of quadrics and optimality issuesoptimality issues

Dupont, Lazard father and son, Petitjean (LORIA and LIP6)

JGA’03

Main result (SoCG’03)Main result (SoCG’03)

Input: 2 implicit quadrics, rational coefficients

near-optimaloptimal

1 extra rational point on conic

rational point on a degree-8 surface

Output: exact parametric representation of intersection Worse-case optimal and always near-optimal

in the degree of the field extension on which the coefficients are defined

Outline of improvementsOutline of improvements

Starting point: Levin’s pencil method

“Simple” (but substantial) improvements: projective setting Gauss reduction of quadratic forms nice parameterizations of projective quadrics

More “involved” improvements: quadric through a rational point avoid explicit reduction ad hoc algorithm for every case

Worth it ??

Levin (1976, 1979)Levin (1976, 1979)

Input: and Find a (simple) ruled quadric in Find frame where is canonical

P transformation matrix, X parameterization Solve degree 2 equation Plug result in X and compute PX

Nested (ruled quadric)Nested (eigenvectors)another (parameterization)

TS QQ

01 PXSPX T

SQ TQRQ

RQ

nestingdepth 5 !!

““Improved projective Levin”Improved projective Levin”

Recall that: Euclidean projective projective quadrics are characterized by

inertia ruled quadrics are those of inertia (3,1) Gauss reduction is rational optimal parameterization of (2,2):

““Improved projective Levin”: Improved projective Levin”: algoalgo

Input: and projective quadrics

Find a quadric of inertia (2,2) in

Find frame where is canonical (Gauss) P transformation matrix, X parameterization

Solve degree 2 equation

Plug result in X and compute PX

SQ TQ

RQ TS QQ RQ

0PXSPX TT

““Improved projective Improved projective Levin”Levin”

Result is in:

Height of the coefficients: d: input quadrics height of output is

ddd 1021768

)()( )( 413 uuPuP

More improvments (generic More improvments (generic case)case)

Features: find a quadric of inertia (2,2) in the pencil

going through a rational point

do not use Gauss directly (but use the rational point found above)

optimize transformations

Near-optimal Near-optimal algorithmalgorithm

Result is in

Height: d: input quadrics, p: rational point height of output is

Observed heights: p = 0 (uncorrelated) random data [-1000,+1000]: 20d (gcd) real data: 5d

52p42d)128(3829 pdpd

)()( )( 413 uuPuP

Near-optimal output: exampleNear-optimal output: example Example coming from real

application (modeling of a tea pot with SGDLsoft)

You could even be lucky!You could even be lucky!

Kill two square roots with one stone…

Can we go further? Can we go further?

In the smooth quartic case, asserting whether the is needed amounts to

finding a rational quadric of the pencil going througha rational point and whose determinant is a square

1. Find a rational point on hyperelliptic curve:

2. Find a rational point on associated quadric:

Rational points on curves and Rational points on curves and surfacessurfaces

ratpoints by Michael Stoll (Univ. of Bremen)

Generically, no solution (computational number theory) confirmed by experiments (random data) but… for real data… different story trivial solutions (rational solutions of f) still valid, but p correlated to d?

52p42d)128(3829 pdpd

Rational point on cone or conicRational point on cone or conic

Theory (Hasse) Q has solutions over the integers iff it

has solutions over the reals and mod every m (Legendre) Q has solutions modulo m for

every m not dividing 2abc (Holzer)

0: 222 czbyaxQ

||||,||||,|||| abzacybcx

Practice Efficient implementation (Cremona & Rusin,

2003) Modular solvers (msolve in Maple)

Re: Example from real dataRe: Example from real data

has canonical equationSQ 0222 zyx

[0,1,1,0] local [-15,4,1,1] global

,151215 22 vuvux ,124 22 vuy

,22 vuw ,z

432234 1052494280407 vuvvuvuu

ConclusionConclusion

Going for near-optimality is important simpler parameterizations smaller height of coefficients

Going for “full optimality” might be important for real data especially when the pencil has rational cones

More experiments needed