On Collins Modular Algorithm for Computing Resultant

1
On Collins Modular Algorithm for Computing
Resultant

• CS874 Course Project
• Yuzhen Xie
• Instructed by Marc Mareno Maza
• April 30, 2003

2
Overview

• Goals of the project
• Significance of the resultant
• Principles of modular computations in Euclidean
  domains
• Principles of modular resultant algorithm of
  Collins
• Implementation in Aldor
• Experimentation and results
• Annex

3
Goals of the Project

• To understand the modular resultant algorithm
• To have an efficient implementation of the
  resultant of two bivariate polynomials
• To experiment the advantage of modular techniques
  in computer algebra
• To learn generic programming with Aldor

4
Significance of the resultant

• Given two polynomials f(v) , g(v)
  , the resultant of f, g is defined
  as the determinant of the Sylvester matrix
• det(S) 0 if and only if f and g have a common
  divisor of positive degree.
• So, if f, g ?(Zu)v then res(f,g) Zu
  and give the values of u for which f, g have
  common roots

n
m
5
Principles of modular computations

• To keep under control the size of the
  coefficients
• fp Z ? Z/pZ is a ring homomorphism.
• That is,
• fp(ab) fp(a) fp(b) , and
• fp(a b) fp(a) fp(b)
• Let A be a square matrix over Z, since det(A) is
  made from additions and multiplications of Z,
  fp(det(A)) det(fp(A)).
• if det(A) lt , then fp(det(A)) det(A).
  
• Therefore, det(fp(A)) det(A)

6
Principles of modular computations (contd)

• For this to work, we need a bound. We have the
  Hadamard bound
• To use Small Primes and machine arithmetic, we
  can use several modular reductions, ?p1, , ?pn
  and recombine the results with the CRA (Chinese
  Remaindering Algorithm)

7
The Modular Resultant Algorithm of Collins (in a
case of two bivariate polynomials)
f(u)(v), g(u)(v) and prime p p1, , pn such
that p does not divide gcd(lc(f), lc(g)), and
p1pn gt Coefficient bound C
?p(f(u)(v)) and ?p(g(u)(v))
Degree bound k, and a a1, , ak1 (ai ?Z/p, and
lc(fp), lc(gp) ? 0 at u a)
?p(f(ua)(v)) and ?p(g(ua)(v))
Compute the resultant r of ?p(f(ua)(v)) and
?p(g(ua)(v))
r1, , rk1 (ri ?Z/p)
Compute the resultant of ?p(f(u)(v)) and
?p(g(u)(v)) by CRA
rp1(u), , rpn(u) (rp1(u) ?Z/p1u, ,
rpn(u)?Z/pnu )
Compute the resultant of f(u)(v) and g(u)(v) by
CRA
8
The Modular Resultant Algorithm of Collins
(contd)

• Coefficient bound
• 2(mn)!dnem, where d max norm (fi) and e max
  norm (gi)
• Degree bound
• degreeu(resv(f.g)) lt degu(f)degv(g)
  degu(g)degv(f)

9
Implementation in Aldor (1)

• //This package computes bounds and helps
  choosing primes
• // Z Integer, M MachineInteger
• BivariateUtilitiesPackage(U UnivariatePolynomialC
  ategory(Z), _
• V UnivariatePolynomialC
  ategory(U)) with
• resultantCoefficientBound (V, V) -gt Z
• resultantDegreeBound (V, V) -gt Z
• primeBad? (V, V, M) -gt Boolean

10
Implementation in Aldor (2)

• // We compute resultant of biv. poly. over a Z/pZ
• ResultantOfBivariatePolynomialsOverSmallPrimeField
  (
• Kp SmallPrimeFieldCategory,
• Up UnivariatePolynomialCategory(Kp),
• Vp UnivariatePolynomialCategory(Up)) with
• evaluationResultant (Vp, Vp, Z) -gt Up
• // the third argument is the degree bound
• // After evaluation, we use the generic
  algorithm
• // for resultants in Z/pZv
• // Then we interpolate them with CRA
• evaluationReduction (Vp, Kp) -gt Up

11
Implementation in Aldor (3)

• ModularResultantOfBivariatePolynomials(
• U UnivariatePolynomialCategory(Z),
• V UnivariatePolynomialCategory(U)) with
• modularResultant (V, V, algorithm Z 1) -gt
  U
• // Top level function
• combine (DUP(Z), Z, DUP(M), M) -gt DUP(Z)
• // Use the CRA for recovering the coefficients
• // the resultant
• resultant (V, V, M, Z, algorithm Z
  1) -gt DUPM
• // Performs the reduction and calls the previous
• // package

12
Experimentation and results

• CRA-CRA modular resultant algorithm of Collins
• Generic essentially the computation of the
  determinate of the Sylvester matrix, without
  taking into account the properties of coefficient
  ring.
• CRA-Generic we use modular computations for the
  coefficients, but the generic method in
  Z/pZuv.
• Experimentation on a PC Pentium IV 2 Gig Hz
• Results for polynomials with very large
  degrees
• Degree of u, v Resultant degree CRA-CRA
  CRA-Generic Generic
• (10,30) 600 87
  352 492
  
• (20,25) 1000 136
  462 635
• (20,50) 2000 1444
  ? gt 2h
• (30,50) 3000 2780
  ? ?

13
Conclusion Remarks

• Now we can compute resultants of polynomials with
  larger degrees in libalgebra (Aldor) than before.
• Consequently we should be able to solve larger
  polynomial systems.
• Our experimentation shows that a mixed method
  (CRA-Generic) is interesting for resultants of
  polynomials with medium degree.