use crate::domains::{EuclideanDomain, Field};
use super::univariate::UnivariatePolynomial;
impl<F: EuclideanDomain> UnivariatePolynomial<F> {
/// Compute the resultant using Brown's polynomial remainder sequence algorithm.
pub fn resultant_prs(&self, other: &Self) -> F::Element {
if self.degree() < other.degree() {
if self.degree() % 2 == 1 && other.degree() % 2 == 1 {
return self.ring.neg(&other.resultant_prs(self));
} else {
return other.resultant_prs(self);
}
}
if other.is_constant() {
return self.ring.pow(&other.get_constant(), self.degree() as u64);
}
let mut a = self.clone();
let mut a_new = other.clone();
let mut deg = a.degree() as u64 - a_new.degree() as u64;
let mut neg_lc = self.ring.one(); //unused
let mut init = false;
let mut beta = self.ring.pow(&self.ring.neg(&self.ring.one()), deg + 1);
let mut psi = self.ring.neg(&self.ring.one());
let mut lcs = vec![(a.lcoeff(), a.degree() as u64)];
while !a_new.is_constant() {
if init {
psi = if deg == 0 {
// can only happen on the first iteration
psi
} else if deg == 1 {
neg_lc.clone()
} else {
let a = self.ring.pow(&neg_lc, deg);
let psi_old = self.ring.pow(&psi, deg - 1);
let (q, r) = self.ring.quot_rem(&a, &psi_old);
debug_assert!(self.ring.is_zero(&r));
q
};
deg = a.degree() as u64 - a_new.degree() as u64;
beta = self.ring.mul(&neg_lc, &self.ring.pow(&psi, deg));
} else {
init = true;
}
neg_lc = self.ring.neg(a_new.coefficients.last().unwrap());
let (_, mut r) = a
.mul_coeff(&self.ring.pow(&neg_lc, deg + 1))
.quot_rem(&a_new);
if (deg + 1) % 2 == 1 {
r = -r;
}
lcs.push((a_new.lcoeff(), a_new.degree() as u64));
(a, a_new) = (a_new, r.div_coeff(&beta));
}
lcs.push((a_new.lcoeff(), 0));
if a_new.is_zero() {
return self.ring.zero();
}
// compute the resultant from the PRS, using the fundamental theorem
let mut rho = self.ring.one();
let mut den = self.ring.one();
for k in 1..lcs.len() {
let mut deg = lcs[k - 1].1 as i64 - lcs[k].1 as i64;
for l in k..lcs.len() - 1 {
deg *= 1 - (lcs[l].1 as i64 - lcs[l + 1].1 as i64);
}
if deg > 0 {
self.ring
.mul_assign(&mut rho, &self.ring.pow(&lcs[k].0, deg as u64));
} else if deg < 0 {
self.ring
.mul_assign(&mut den, &self.ring.pow(&lcs[k].0, (-deg) as u64));
}
}
self.ring.quot_rem(&rho, &den).0
}
/// Compute the resultant using a primitive polynomial remainder sequence.
pub fn resultant_primitive(&self, other: &Self) -> F::Element {
if self.degree() < other.degree() {
if self.degree() % 2 == 1 && other.degree() % 2 == 1 {
return self.ring.neg(&other.resultant_primitive(self));
} else {
return other.resultant_primitive(self);
}
}
let mut a = self.clone();
let mut a_new = other.clone();
let mut v = vec![a.degree()];
let mut c = vec![a.lcoeff()];
let mut ab = vec![(self.ring.one(), self.ring.one())];
while !a_new.is_constant() {
let n = a.degree() as u64 + 1 - a_new.degree() as u64;
let alpha = self.ring.pow(&a_new.lcoeff(), n);
let (_, mut r) = a.clone().mul_coeff(&alpha).quot_rem(&a_new);
let beta = r.content();
r = r.div_coeff(&beta);
(a, a_new) = (a_new, r);
v.push(a.degree());
c.push(a.lcoeff());
ab.push((alpha, beta));
}
let r = a_new.lcoeff();
if self.ring.is_zero(&r) {
return r;
}
let mut sign = 0;
for w in v.windows(2) {
sign += w[0] * w[1];
}
let mut res = self.ring.pow(&r, *v.last().unwrap() as u64);
if sign % 2 == 1 {
res = self.ring.neg(&res);
};
v.push(0);
let mut num = self.ring.one();
let mut den = self.ring.one();
for i in 1..c.len() {
self.ring.mul_assign(
&mut res,
&self.ring.pow(&c[i], v[i - 1] as u64 - v[i + 1] as u64),
);
self.ring
.mul_assign(&mut num, &self.ring.pow(&ab[i].1, v[i] as u64));
self.ring
.mul_assign(&mut den, &self.ring.pow(&ab[i].0, v[i] as u64));
}
let (q, r) = self.ring.quot_rem(&self.ring.mul(&num, &res), &den);
assert!(self.ring.is_zero(&r));
q
}
}
impl<F: Field> UnivariatePolynomial<F> {
/// Compute the resultant of the two polynomials.
pub fn resultant(&self, other: &Self) -> F::Element {
let mut a = self.clone();
let mut a_new = other.clone();
let mut v = vec![a.degree()];
let mut c = vec![a.lcoeff()];
while !a_new.is_constant() {
let (_, r) = a.quot_rem(&a_new);
(a, a_new) = (a_new, r);
v.push(a.degree());
c.push(a.lcoeff());
}
let r = a_new.lcoeff();
if self.ring.is_zero(&r) {
return r;
}
let mut sign = 0;
for w in v.windows(2) {
sign += w[0] * w[1];
}
let mut res = self.ring.pow(&r, *v.last().unwrap() as u64);
if sign % 2 == 1 {
res = self.ring.neg(&res);
};
v.push(0);
for i in 1..c.len() {
self.ring.mul_assign(
&mut res,
&self.ring.pow(&c[i], v[i - 1] as u64 - v[i + 1] as u64),
);
}
res
}
}
#[cfg(test)]
mod test {
use std::sync::Arc;
use crate::atom::AtomCore;
use crate::domains::integer::Z;
use crate::domains::rational::Q;
use crate::domains::rational_polynomial::{
FromNumeratorAndDenominator, RationalPolynomial, RationalPolynomialField,
};
use crate::poly::polynomial::MultivariatePolynomial;
use crate::{parse, symbol};
#[test]
fn resultant() {
let a = parse!("9v1^6-27v1^4-27v1^3+72v1^2+18v1-451")
.to_polynomial::<_, u8>(&Q, None)
.to_univariate_from_univariate(0);
let b = parse!("3v1^4-4v1^2-9v1+21")
.to_polynomial::<_, u8>(&Q, None)
.to_univariate_from_univariate(0);
let r = a.resultant(&b);
assert_eq!(r, 11149673028381u64);
}
#[test]
fn res_methods() {
let (x, y, z) = symbol!("v1", "v2", "v3");
let vars = Arc::new(vec![x.into(), y.into(), z.into()]);
let a = parse!("2v1^2").to_polynomial::<_, u8>(&Z, Some(vars.clone()));
let aa = a.to_univariate(0).map_coeff(
|x| RationalPolynomial::from_num_den(x.clone(), x.one(), &x.ring, false),
RationalPolynomialField::from_poly(&a),
);
let b = parse!("v3^2+v1^5")
.to_polynomial::<_, u8>(&Z, Some(vars.clone()))
.to_univariate(0)
.map_coeff(
|x| RationalPolynomial::from_num_den(x.clone(), x.one(), &x.ring, false),
RationalPolynomialField::from_poly(&a),
);
let prs = aa.resultant_prs(&b);
let prim = aa.resultant_primitive(&b);
let res = aa.resultant(&b);
assert_eq!(prs, prim);
assert_eq!(prim, res);
}
#[test]
fn resultant_prs_large() {
let system = [
"-272853213601 + 114339252960*v2 - 4841413740*v2^2 + 296664007680*v4 - 25123011840*v2*v4 -
32592015360*v4^2 - 4907531205*v5 + 6155208630*v5^2 - 3860046090*v5^3 + 1210353435*v5^4 -
151807041*v5^5 + 312814245280*v6 - 97612876080*v2*v6 + 1518070410*v2^2*v6 -
253265840640*v4*v6 + 7877554560*v2*v4*v6 + 10219530240*v4^2*v6 - 146051082720*v6^2 +
29048482440*v2*v6^2 + 75369035520*v4*v6^2 + 35852138640*v6^3 - 3036140820*v2*v6^3 -
7877554560*v4*v6^3 - 4841413740*v6^4 + 303614082*v6^5",
"-121828703201 - 1128406464*v1 + 303614082*v1^2 + 24547177584*v2 - 303614082*v2^2 -
2927757312*v3 + 1575510912*v1*v3 + 2043906048*v3^2 + 123022775808*v4 - 6600113280*v2*v4 -
15080712192*v4^2 + 1480577211*v5 + 146055798*v5^2 - 1347744906*v5^3 + 816475707*v5^4 -
151807041*v5^5 + 171636450272*v6 - 32717479104*v2*v6 + 303614082*v2^2*v6 -
135541762560*v4*v6 + 3151021824*v2*v4*v6 + 6131718144*v4^2*v6 - 95005523376*v6^2 +
13441077468*v2*v6^2 + 49947954048*v4*v6^2 + 26959925088*v6^3 - 1821684492*v2*v6^3 -
6302043648*v4*v6^3 - 4176745074*v6^4 + 303614082*v6^5",
];
let mut system = system
.iter()
.map(|s| parse!(s).to_polynomial::<_, u16>(&Z, None))
.collect::<Vec<_>>();
MultivariatePolynomial::unify_variables_list(&mut system);
let var = 0;
let a = system[0].to_univariate(var);
let b = system[1].to_univariate(var);
let r = a.resultant_prs(&b);
let res = 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let res = parse!(res).to_polynomial::<_, u16>(&Z, system[0].variables.clone());
assert_eq!(r, res);
}
}