/**
* GenPolynomial absolute factorization of a irreducible polynomial.
*
* @param P irreducible! GenPolynomial.
* @return factors container: [p_1,...,p_k] with P = prod_{i=1, ..., k} p_i in K(alpha)[x] for
* suitable alpha and p_i irreducible over L[x], where K \subset K(alpha) \subset L is an
* algebraically closed field over K. <b>Note:</b> K(alpha) not yet minimal.
*/
public Factors<C> factorsAbsoluteIrreducible(GenPolynomial<C> P) {
if (P == null) {
throw new RuntimeException(this.getClass().getName() + " P == null");
}
if (P.isZERO()) {
return new Factors<C>(P);
}
GenPolynomialRing<C> pfac = P.ring; // K[x]
if (pfac.nvar <= 1) {
return baseFactorsAbsoluteIrreducible(P);
}
if (!pfac.coFac.isField()) {
throw new RuntimeException("only for field coefficients");
}
List<GenPolynomial<C>> factors = new ArrayList<GenPolynomial<C>>();
if (P.degree() <= 1) {
return new Factors<C>(P);
}
// find field extension K(alpha)
GenPolynomial<C> up = P;
RingFactory<C> cf = pfac.coFac;
long cr = cf.characteristic().longValue(); // char might be larger
if (cr == 0L) {
cr = Long.MAX_VALUE;
}
long rp = 0L;
for (int i = 0; i < (pfac.nvar - 1); i++) {
rp = 0L;
GenPolynomialRing<C> nfac = pfac.contract(1);
String[] vn = new String[] {pfac.getVars()[pfac.nvar - 1]};
GenPolynomialRing<GenPolynomial<C>> rfac =
new GenPolynomialRing<GenPolynomial<C>>(nfac, 1, pfac.tord, vn);
GenPolynomial<GenPolynomial<C>> upr = PolyUtil.<C>recursive(rfac, up);
// System.out.println("upr = " + upr);
GenPolynomial<C> ep;
do {
if (rp >= cr) {
throw new RuntimeException("elements of prime field exhausted: " + cr);
}
C r = cf.fromInteger(rp); // cf.random(rp);
// System.out.println("r = " + r);
ep = PolyUtil.<C>evaluateMain(nfac, upr, r);
// System.out.println("ep = " + ep);
rp++;
} while (!isSquarefree(ep) /*todo: || ep.degree() <= 1*/); // max deg
up = ep;
pfac = nfac;
}
up = up.monic();
if (debug) {
logger.info("P(" + rp + ") = " + up);
// System.out.println("up = " + up);
}
if (debug && !isSquarefree(up)) {
throw new RuntimeException("not irreducible up = " + up);
}
if (up.degree(0) <= 1) {
return new Factors<C>(P);
}
// find irreducible factor of up
List<GenPolynomial<C>> UF = baseFactorsSquarefree(up);
// System.out.println("UF = " + UF);
FactorsList<C> aUF = baseFactorsAbsoluteSquarefree(up);
// System.out.println("aUF = " + aUF);
AlgebraicNumberRing<C> arfac = aUF.findExtensionField();
// System.out.println("arfac = " + arfac);
long e = up.degree(0);
// search factor polynomial with smallest degree
for (int i = 0; i < UF.size(); i++) {
GenPolynomial<C> upi = UF.get(i);
long d = upi.degree(0);
if (1 <= d && d <= e) {
up = upi;
e = up.degree(0);
}
}
if (up.degree(0) <= 1) {
return new Factors<C>(P);
}
if (debug) {
logger.info("field extension by " + up);
}
List<GenPolynomial<AlgebraicNumber<C>>> afactors =
new ArrayList<GenPolynomial<AlgebraicNumber<C>>>();
// setup field extension K(alpha)
// String[] vars = new String[] { "z_" + Math.abs(up.hashCode() % 1000) };
String[] vars = pfac.newVars("z_");
pfac = pfac.clone();
String[] ovars = pfac.setVars(vars); // side effects!
GenPolynomial<C> aup = pfac.copy(up); // hack to exchange the variables
// AlgebraicNumberRing<C> afac = new AlgebraicNumberRing<C>(aup,true); // since irreducible
AlgebraicNumberRing<C> afac = arfac;
int depth = afac.depth();
// System.out.println("afac = " + afac);
GenPolynomialRing<AlgebraicNumber<C>> pafac =
new GenPolynomialRing<AlgebraicNumber<C>>(afac, P.ring.nvar, P.ring.tord, P.ring.getVars());
// System.out.println("pafac = " + pafac);
// convert to K(alpha)
GenPolynomial<AlgebraicNumber<C>> Pa =
PolyUtil.<C>convertToRecAlgebraicCoefficients(depth, pafac, P);
// System.out.println("Pa = " + Pa);
// factor over K(alpha)
FactorAbstract<AlgebraicNumber<C>> engine = FactorFactory.<C>getImplementation(afac);
afactors = engine.factorsSquarefree(Pa);
if (debug) {
logger.info("K(alpha) factors multi = " + afactors);
// System.out.println("K(alpha) factors = " + afactors);
}
if (afactors.size() <= 1) {
return new Factors<C>(P);
}
// normalize first factor to monic
GenPolynomial<AlgebraicNumber<C>> p1 = afactors.get(0);
AlgebraicNumber<C> p1c = p1.leadingBaseCoefficient();
if (!p1c.isONE()) {
GenPolynomial<AlgebraicNumber<C>> p2 = afactors.get(1);
afactors.remove(p1);
afactors.remove(p2);
p1 = p1.divide(p1c);
p2 = p2.multiply(p1c);
afactors.add(p1);
afactors.add(p2);
}
// recursion for splitting field
// find minimal field extension K(beta) \subset K(alpha)
return new Factors<C>(P, afac, Pa, afactors);
}
/**
* GenPolynomial base absolute factorization of a irreducible polynomial.
*
* @param P irreducible! univariate GenPolynomial.
* @return factors container: [p_1,...,p_k] with P = prod_{i=1, ..., k} p_i in K(alpha)[x] for
* suitable alpha and p_i irreducible over L[x], where K \subset K(alpha) \subset L is an
* algebraically closed field over K. <b>Note:</b> K(alpha) not yet minimal.
*/
public Factors<C> baseFactorsAbsoluteIrreducible(GenPolynomial<C> P) {
if (P == null) {
throw new RuntimeException(this.getClass().getName() + " P == null");
}
if (P.isZERO()) {
return new Factors<C>(P);
}
// System.out.println("\nP_base_irred = " + P);
GenPolynomialRing<C> pfac = P.ring; // K[x]
if (pfac.nvar > 1) {
// System.out.println("facs_base_irred: univ");
throw new RuntimeException("only for univariate polynomials");
}
if (!pfac.coFac.isField()) {
// System.out.println("facs_base_irred: field");
throw new RuntimeException("only for field coefficients");
}
if (P.degree(0) <= 1) {
return new Factors<C>(P);
}
// setup field extension K(alpha) where alpha = z_xx
// String[] vars = new String[] { "z_" + Math.abs(P.hashCode() % 1000) };
String[] vars = pfac.newVars("z_");
pfac = pfac.clone();
vars = pfac.setVars(vars);
GenPolynomial<C> aP = pfac.copy(P); // hack to exchange the variables
AlgebraicNumberRing<C> afac = new AlgebraicNumberRing<C>(aP, true); // since irreducible
if (logger.isInfoEnabled()) {
logger.info("K(alpha) = " + afac);
logger.info("K(alpha) = " + afac.toScript());
// System.out.println("K(alpha) = " + afac);
}
GenPolynomialRing<AlgebraicNumber<C>> pafac =
new GenPolynomialRing<AlgebraicNumber<C>>(afac, aP.ring.nvar, aP.ring.tord, /*old*/ vars);
// convert to K(alpha)
GenPolynomial<AlgebraicNumber<C>> Pa = PolyUtil.<C>convertToAlgebraicCoefficients(pafac, P);
if (logger.isInfoEnabled()) {
logger.info("P over K(alpha) = " + Pa);
// logger.info("P over K(alpha) = " + Pa.toScript());
// System.out.println("P in K(alpha) = " + Pa);
}
// factor over K(alpha)
FactorAbstract<AlgebraicNumber<C>> engine = FactorFactory.<C>getImplementation(afac);
// System.out.println("K(alpha) engine = " + engine);
List<GenPolynomial<AlgebraicNumber<C>>> factors = engine.baseFactorsSquarefree(Pa);
// System.out.println("factors = " + factors);
if (logger.isInfoEnabled()) {
logger.info("factors over K(alpha) = " + factors);
// System.out.println("factors over K(alpha) = " + factors);
}
List<GenPolynomial<AlgebraicNumber<C>>> faca =
new ArrayList<GenPolynomial<AlgebraicNumber<C>>>(factors.size());
;
List<Factors<AlgebraicNumber<C>>> facar = new ArrayList<Factors<AlgebraicNumber<C>>>();
for (GenPolynomial<AlgebraicNumber<C>> fi : factors) {
if (fi.degree(0) <= 1) {
faca.add(fi);
} else {
// System.out.println("fi.deg > 1 = " + fi);
FactorAbsolute<AlgebraicNumber<C>> aengine =
(FactorAbsolute<AlgebraicNumber<C>>) FactorFactory.<C>getImplementation(afac);
Factors<AlgebraicNumber<C>> fif = aengine.baseFactorsAbsoluteIrreducible(fi);
// System.out.println("fif = " + fif);
facar.add(fif);
}
}
if (facar.size() == 0) {
facar = null;
}
// find minimal field extension K(beta) \subset K(alpha)
return new Factors<C>(P, afac, Pa, faca, facar);
}
