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/*
* $Id: SquarefreeInfiniteAlgebraicFieldCharP.java 3290 2010-08-26 09:18:48Z
* kredel $
*/
package edu.jas.ufd;
import java.util.ArrayList;
import java.util.List;
import java.util.Map;
import java.util.SortedMap;
import java.util.TreeMap;
import org.apache.log4j.Logger;
import edu.jas.gb.GroebnerBaseAbstract;
import edu.jas.gb.GroebnerBaseSeq;
import edu.jas.gb.Reduction;
import edu.jas.gb.ReductionSeq;
import edu.jas.poly.AlgebraicNumber;
import edu.jas.poly.AlgebraicNumberRing;
import edu.jas.poly.ExpVector;
import edu.jas.poly.GenPolynomial;
import edu.jas.poly.GenPolynomialRing;
import edu.jas.poly.Monomial;
import edu.jas.poly.PolyUtil;
import edu.jas.structure.GcdRingElem;
import edu.jas.structure.Power;
import edu.jas.structure.RingFactory;
/**
* Squarefree decomposition for algebraic extensions of infinite coefficient
* fields of characteristic p > 0.
* @author Heinz Kredel
*/
public class SquarefreeInfiniteAlgebraicFieldCharP> extends
SquarefreeFieldCharP> {
private static final Logger logger = Logger.getLogger(SquarefreeInfiniteAlgebraicFieldCharP.class);
//private final boolean debug = logger.isDebugEnabled();
/**
* Squarefree engine for infinite ring of characteristic p base coefficients.
*/
protected final SquarefreeAbstract aengine;
/**
* Constructor.
*/
public SquarefreeInfiniteAlgebraicFieldCharP(RingFactory> fac) {
super(fac);
// isFinite() predicate now present
if (fac.isFinite()) {
throw new IllegalArgumentException("fac must be in-finite");
}
AlgebraicNumberRing afac = (AlgebraicNumberRing) fac;
GenPolynomialRing rfac = afac.ring;
//System.out.println("rfac = " + rfac);
//System.out.println("rfac = " + rfac.coFac);
aengine = SquarefreeFactory. getImplementation(rfac);
//System.out.println("aengine = " + aengine);
}
/* --------- algebraic number char-th roots --------------------- */
/**
* Squarefree factors of a AlgebraicNumber.
* @param P AlgebraicNumber.
* @return [p_1 -> e_1,...,p_k - > e_k] with P = prod_{i=1, ..., k}
* p_i**e_k.
*/
@Override
public SortedMap, Long> squarefreeFactors(AlgebraicNumber P) {
if (P == null) {
throw new IllegalArgumentException(this.getClass().getName() + " P == null");
}
SortedMap, Long> factors = new TreeMap, Long>();
if (P.isZERO()) {
return factors;
}
if (P.isONE()) {
factors.put(P, 1L);
return factors;
}
GenPolynomial an = P.val;
AlgebraicNumberRing pfac = P.ring;
if (!an.isONE()) {
//System.out.println("an = " + an);
//System.out.println("aengine = " + aengine);
SortedMap, Long> nfac = aengine.squarefreeFactors(an);
//System.out.println("nfac = " + nfac);
for (Map.Entry, Long> me : nfac.entrySet()) {
GenPolynomial nfp = me.getKey();
AlgebraicNumber nf = new AlgebraicNumber(pfac, nfp);
factors.put(nf, me.getValue()); //nfac.get(nfp));
}
}
if (factors.size() == 0) {
factors.put(P, 1L);
}
return factors;
}
/**
* Characteristics root of a AlgebraicNumber.
* @param P AlgebraicNumber.
* @return [p -> k] if exists k with e=charactristic(P)*k and P = p**e,
* else null.
*/
public SortedMap, Long> rootCharacteristic(AlgebraicNumber P) {
if (P == null) {
throw new IllegalArgumentException(this.getClass().getName() + " P == null");
}
java.math.BigInteger c = P.ring.characteristic();
if (c.signum() == 0) {
return null;
}
SortedMap, Long> root = new TreeMap, Long>();
if (P.isZERO()) {
return root;
}
if (P.isONE()) {
root.put(P, 1L);
return root;
}
// generate system of equations
AlgebraicNumberRing afac = P.ring;
long deg = afac.modul.degree(0);
int d = (int) deg;
String[] vn = GenPolynomialRing.newVars("c", d);
GenPolynomialRing> pfac = new GenPolynomialRing>(afac, d, vn);
List>> uv = (List>>) pfac
.univariateList();
GenPolynomial> cp = pfac.getZERO();
GenPolynomialRing apfac = afac.ring;
long i = 0;
for (GenPolynomial> pa : uv) {
GenPolynomial ca = apfac.univariate(0, i++);
GenPolynomial> pb = pa.multiply(new AlgebraicNumber(afac, ca));
cp = cp.sum(pb);
}
GenPolynomial> cpp = Power
.>> positivePower(cp, c);
if (logger.isInfoEnabled()) {
logger.info("cp = " + cp);
logger.info("cp^p = " + cpp);
logger.info("P = " + P);
}
GenPolynomialRing ppfac = new GenPolynomialRing(apfac.coFac, pfac);
List> gl = new ArrayList>();
if (deg == c.longValue() && afac.modul.length() == 2) {
logger.info("deg(" + deg + ") == char_p(" + c.longValue() + ")");
for (Monomial> m : cpp) {
ExpVector f = m.e;
AlgebraicNumber a = m.c;
//System.out.println("a = " + a + " : " + a.toScriptFactory());
GenPolynomial ap = a.val;
for (Monomial ma : ap) {
ExpVector e = ma.e;
C cc = ma.c;
C pc = P.val.coefficient(e);
C cc1 = ((RingFactory) pc.factory()).getONE();
C pc1 = ((RingFactory) pc.factory()).getZERO();
//System.out.println("cc = " + cc + ", e = " + e);
//System.out.println("pc = " + pc + " : " + cc.toScriptFactory());
if (cc instanceof AlgebraicNumber && pc instanceof AlgebraicNumber) {
throw new UnsupportedOperationException(
"case multiple algebraic extensions not implemented");
} else if (cc instanceof Quotient && pc instanceof Quotient) {
Quotient ccp = (Quotient) (Object) cc;
Quotient pcp = (Quotient) (Object) pc;
if (pcp.isConstant()) {
//logger.error("finite field not allowed here " + afac.toScript());
throw new ArithmeticException("finite field not allowed here " + afac.toScript());
}
//C dc = cc.divide(pc);
Quotient dcp = ccp.divide(pcp);
if (dcp.isConstant()) { // not possible: dc.isConstant()
//System.out.println("dcp = " + dcp + " : " + cc.toScriptFactory()); // + ", dc = " + dc);
//if ( dcp.num.isConstant() )
cc1 = cc;
pc1 = pc;
}
GenPolynomial r = new GenPolynomial(ppfac, cc1, f);
r = r.subtract(pc1);
//System.out.println("r = " + r);
gl.add(r);
}
}
}
} else {
for (Monomial> m : cpp) {
ExpVector f = m.e;
AlgebraicNumber a = m.c;
//System.out.println("m = " + m);
GenPolynomial ap = a.val;
for (Monomial ma : ap) {
ExpVector e = ma.e;
C cc = ma.c;
C pc = P.val.coefficient(e);
GenPolynomial r = new GenPolynomial(ppfac, cc, f);
r = r.subtract(pc);
//System.out.println("r = " + r);
gl.add(r);
}
}
}
if (logger.isInfoEnabled()) {
logger.info("equations = " + gl);
}
// solve system of equations and construct result
Reduction red = new ReductionSeq();
gl = red.irreducibleSet(gl);
GroebnerBaseAbstract bb = new GroebnerBaseSeq(); //GBFactory.getImplementation();
int z = bb.commonZeroTest(gl);
if (z < 0) { // no solution
return null;
}
if (logger.isInfoEnabled()) {
logger.info("solution = " + gl);
}
GenPolynomial car = apfac.getZERO();
for (GenPolynomial pl : gl) {
if (pl.length() <= 1) {
continue;
}
if (pl.length() > 2) {
throw new IllegalArgumentException("dim > 0 not implemented " + pl);
}
//System.out.println("pl = " + pl);
ExpVector e = pl.leadingExpVector();
int[] v = e.dependencyOnVariables();
if (v == null || v.length == 0) {
continue;
}
int vi = v[0];
//System.out.println("vi = " + vi);
GenPolynomial ca = apfac.univariate(0, deg - 1 - vi);
//System.out.println("ca = " + ca);
C tc = pl.trailingBaseCoefficient();
tc = tc.negate();
if (e.maxDeg() == c.longValue()) { // p-th root of tc ...
//SortedMap br = aengine.rootCharacteristic(tc);
SortedMap br = aengine.squarefreeFactors(tc);
//System.out.println("br = " + br);
if (br != null && br.size() > 0) {
C cc = apfac.coFac.getONE();
for (Map.Entry me : br.entrySet()) {
C bc = me.getKey();
long ll = me.getValue(); //br.get(bc);
if (ll % c.longValue() == 0L) {
long fl = ll / c.longValue();
cc = cc.multiply(Power. positivePower(bc, fl));
} else { // fail ?
cc = cc.multiply(bc);
}
}
//System.out.println("cc = " + cc);
tc = cc;
}
}
ca = ca.multiply(tc);
car = car.sum(ca);
}
AlgebraicNumber rr = new AlgebraicNumber(afac, car);
if (logger.isInfoEnabled()) {
logger.info("solution AN = " + rr);
//System.out.println("rr = " + rr);
}
root.put(rr, 1L);
return root;
}
/**
* GenPolynomial char-th root main variable.
* @param P univariate GenPolynomial with AlgebraicNumber coefficients.
* @return char-th_rootOf(P), or null, if P is no char-th root.
*/
public GenPolynomial> rootCharacteristic(GenPolynomial> P) {
if (P == null || P.isZERO()) {
return P;
}
GenPolynomialRing> pfac = P.ring;
if (pfac.nvar > 1) {
// go to recursion
GenPolynomialRing> cfac = pfac.contract(1);
GenPolynomialRing>> rfac = new GenPolynomialRing>>(
cfac, 1);
GenPolynomial>> Pr = PolyUtil.> recursive(
rfac, P);
GenPolynomial>> Prc = recursiveUnivariateRootCharacteristic(Pr);
if (Prc == null) {
return null;
}
GenPolynomial> D = PolyUtil.> distribute(pfac, Prc);
return D;
}
RingFactory> rf = pfac.coFac;
if (rf.characteristic().signum() != 1) {
// basePthRoot not possible
throw new IllegalArgumentException(P.getClass().getName() + " only for ModInteger polynomials "
+ rf);
}
long mp = rf.characteristic().longValue();
GenPolynomial> d = pfac.getZERO().copy();
for (Monomial> m : P) {
ExpVector f = m.e;
long fl = f.getVal(0);
if (fl % mp != 0) {
return null;
}
fl = fl / mp;
SortedMap, Long> sm = rootCharacteristic(m.c);
if (sm == null) {
return null;
}
if (logger.isInfoEnabled()) {
logger.info("sm_alg,root = " + sm);
}
AlgebraicNumber r = rf.getONE();
for (Map.Entry, Long> me : sm.entrySet()) {
AlgebraicNumber rp = me.getKey();
long gl = me.getValue(); //sm.get(rp);
if (gl > 1) {
rp = Power.> positivePower(rp, gl);
}
r = r.multiply(rp);
}
ExpVector e = ExpVector.create(1, 0, fl);
d.doPutToMap(e, r);
}
logger.info("sm_alg,root,d = " + d);
return d;
}
/**
* GenPolynomial char-th root univariate polynomial.
* @param P GenPolynomial.
* @return char-th_rootOf(P).
*/
@Override
public GenPolynomial> baseRootCharacteristic(GenPolynomial> P) {
if (P == null || P.isZERO()) {
return P;
}
GenPolynomialRing> pfac = P.ring;
if (pfac.nvar > 1) {
// basePthRoot not possible by return type
throw new IllegalArgumentException(P.getClass().getName() + " only for univariate polynomials");
}
RingFactory> rf = pfac.coFac;
if (rf.characteristic().signum() != 1) {
// basePthRoot not possible
throw new IllegalArgumentException(P.getClass().getName() + " only for char p > 0 " + rf);
}
long mp = rf.characteristic().longValue();
GenPolynomial> d = pfac.getZERO().copy();
for (Monomial> m : P) {
//System.out.println("m = " + m);
ExpVector f = m.e;
long fl = f.getVal(0);
if (fl % mp != 0) {
return null;
}
fl = fl / mp;
SortedMap, Long> sm = rootCharacteristic(m.c);
if (sm == null) {
return null;
}
if (logger.isInfoEnabled()) {
logger.info("sm_alg,base,root = " + sm);
}
AlgebraicNumber r = rf.getONE();
for (Map.Entry, Long> me : sm.entrySet()) {
AlgebraicNumber rp = me.getKey();
//System.out.println("rp = " + rp);
long gl = me.getValue(); //sm.get(rp);
//System.out.println("gl = " + gl);
AlgebraicNumber re = rp;
if (gl > 1) {
re = Power.> positivePower(rp, gl);
}
//System.out.println("re = " + re);
r = r.multiply(re);
}
ExpVector e = ExpVector.create(1, 0, fl);
d.doPutToMap(e, r);
}
if (logger.isInfoEnabled()) {
logger.info("sm_alg,base,d = " + d);
}
return d;
}
/**
* GenPolynomial char-th root univariate polynomial with polynomial
* coefficients.
* @param P recursive univariate GenPolynomial.
* @return char-th_rootOf(P), or null if P is no char-th root.
*/
@Override
public GenPolynomial>> recursiveUnivariateRootCharacteristic(
GenPolynomial>> P) {
if (P == null || P.isZERO()) {
return P;
}
GenPolynomialRing>> pfac = P.ring;
if (pfac.nvar > 1) {
// basePthRoot not possible by return type
throw new IllegalArgumentException(P.getClass().getName()
+ " only for univariate recursive polynomials");
}
RingFactory>> rf = pfac.coFac;
if (rf.characteristic().signum() != 1) {
// basePthRoot not possible
throw new IllegalArgumentException(P.getClass().getName() + " only for char p > 0 " + rf);
}
long mp = rf.characteristic().longValue();
GenPolynomial>> d = pfac.getZERO().copy();
for (Monomial>> m : P) {
ExpVector f = m.e;
long fl = f.getVal(0);
if (fl % mp != 0) {
return null;
}
fl = fl / mp;
GenPolynomial> r = rootCharacteristic(m.c);
if (r == null) {
return null;
}
ExpVector e = ExpVector.create(1, 0, fl);
d.doPutToMap(e, r);
}
return d;
}
}
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