Source code

001/*
002 * $Id: HenselMultUtil.java 6007 2020-03-29 13:34:49Z kredel $
003 */
004
005package edu.jas.ufd;
006
007
008import java.util.ArrayList;
009import java.util.List;
010
011import org.apache.logging.log4j.Logger;
012import org.apache.logging.log4j.LogManager; 
013
014import edu.jas.arith.BigInteger;
015import edu.jas.arith.ModIntegerRing;
016import edu.jas.arith.ModLongRing;
017import edu.jas.arith.Modular;
018import edu.jas.arith.ModularRingFactory;
019import edu.jas.poly.GenPolynomial;
020import edu.jas.poly.GenPolynomialRing;
021import edu.jas.poly.PolyUtil;
022import edu.jas.ps.PolynomialTaylorFunction;
023import edu.jas.ps.TaylorFunction;
024import edu.jas.ps.UnivPowerSeries;
025import edu.jas.ps.UnivPowerSeriesRing;
026import edu.jas.structure.GcdRingElem;
027import edu.jas.structure.RingFactory;
028
029
030/**
031 * Hensel multivariate lifting utilities.
032 * @author Heinz Kredel
033 */
034
035public class HenselMultUtil {
036
037
038    private static final Logger logger = LogManager.getLogger(HenselMultUtil.class);
039
040
041    private static final boolean debug = logger.isInfoEnabled();
042
043
044    /**
045     * Modular diophantine equation solution and lifting algorithm. Let p =
046     * A_i.ring.coFac.modul() and assume ggt(A,B) == 1 mod p.
047     * @param A modular GenPolynomial, mod p^k
048     * @param B modular GenPolynomial, mod p^k
049     * @param C modular GenPolynomial, mod p^k
050     * @param V list of substitution values, mod p^k
051     * @param d desired approximation exponent (x_i-v_i)^d.
052     * @param k desired approximation exponent p^k.
053     * @return [s, t] with s A' + t B' = C mod p^k, with A' = B, B' = A.
054     */
055    public static <MOD extends GcdRingElem<MOD> & Modular> List<GenPolynomial<MOD>> liftDiophant(
056                    GenPolynomial<MOD> A, GenPolynomial<MOD> B, GenPolynomial<MOD> C, List<MOD> V, long d,
057                    long k) throws NoLiftingException {
058        GenPolynomialRing<MOD> pkfac = C.ring;
059        if (pkfac.nvar == 1) { // V, d ignored
060            return HenselUtil.<MOD> liftDiophant(A, B, C, k);
061        }
062        if (!pkfac.equals(A.ring)) {
063            throw new IllegalArgumentException("A.ring != pkfac: " + A.ring + " != " + pkfac);
064        }
065
066        // evaluate at v_n:
067        List<MOD> Vp = new ArrayList<MOD>(V);
068        MOD v = Vp.remove(Vp.size() - 1);
069        //GenPolynomial<MOD> zero = pkfac.getZERO();
070        // (x_n - v)
071        GenPolynomial<MOD> mon = pkfac.getONE();
072        GenPolynomial<MOD> xv = pkfac.univariate(0, 1);
073        xv = xv.subtract(pkfac.fromInteger(v.getSymmetricInteger().getVal()));
074        //System.out.println("xv = " + xv);
075        // A(v), B(v), C(v)
076        ModularRingFactory<MOD> cf = (ModularRingFactory<MOD>) pkfac.coFac;
077        MOD vp = cf.fromInteger(v.getSymmetricInteger().getVal());
078        //System.out.println("v = " + v + ", vp = " + vp);
079        GenPolynomialRing<MOD> ckfac = pkfac.contract(1);
080        GenPolynomial<MOD> Ap = PolyUtil.<MOD> evaluateMain(ckfac, A, vp);
081        GenPolynomial<MOD> Bp = PolyUtil.<MOD> evaluateMain(ckfac, B, vp);
082        GenPolynomial<MOD> Cp = PolyUtil.<MOD> evaluateMain(ckfac, C, vp);
083        //System.out.println("Ap = " + Ap);
084        //System.out.println("Bp = " + Bp);
085        //System.out.println("Cp = " + Cp);
086
087        // recursion:
088        List<GenPolynomial<MOD>> su = HenselMultUtil.<MOD> liftDiophant(Ap, Bp, Cp, Vp, d, k);
089        //System.out.println("su@p^" + k + " = " + su);
090        //System.out.println("coFac = " + su.get(0).ring.coFac.toScript());
091        if (pkfac.nvar == 2 && !HenselUtil.<MOD> isDiophantLift(Bp, Ap, su.get(0), su.get(1), Cp)) {
092            //System.out.println("isDiophantLift: false");
093            throw new NoLiftingException("isDiophantLift: false");
094        }
095        if (!ckfac.equals(su.get(0).ring)) {
096            throw new IllegalArgumentException("qfac != ckfac: " + su.get(0).ring + " != " + ckfac);
097        }
098        GenPolynomialRing<BigInteger> ifac = new GenPolynomialRing<BigInteger>(new BigInteger(), pkfac);
099        //System.out.println("ifac = " + ifac.toScript());
100        String[] mn = new String[] { pkfac.getVars()[pkfac.nvar - 1] };
101        GenPolynomialRing<GenPolynomial<MOD>> qrfac = new GenPolynomialRing<GenPolynomial<MOD>>(ckfac, 1, mn);
102        //System.out.println("qrfac = " + qrfac);
103
104        List<GenPolynomial<MOD>> sup = new ArrayList<GenPolynomial<MOD>>(su.size());
105        List<GenPolynomial<BigInteger>> supi = new ArrayList<GenPolynomial<BigInteger>>(su.size());
106        for (GenPolynomial<MOD> s : su) {
107            GenPolynomial<MOD> sp = s.extend(pkfac, 0, 0L);
108            sup.add(sp);
109            GenPolynomial<BigInteger> spi = PolyUtil.integerFromModularCoefficients(ifac, sp);
110            supi.add(spi);
111        }
112        //System.out.println("sup  = " + sup);
113        //System.out.println("supi = " + supi);
114        GenPolynomial<BigInteger> Ai = PolyUtil.integerFromModularCoefficients(ifac, A);
115        GenPolynomial<BigInteger> Bi = PolyUtil.integerFromModularCoefficients(ifac, B);
116        GenPolynomial<BigInteger> Ci = PolyUtil.integerFromModularCoefficients(ifac, C);
117        //System.out.println("Ai = " + Ai);
118        //System.out.println("Bi = " + Bi);
119        //System.out.println("Ci = " + Ci);
120        //GenPolynomial<MOD> aq = PolyUtil.<MOD> fromIntegerCoefficients(pkfac, Ai);
121        //GenPolynomial<MOD> bq = PolyUtil.<MOD> fromIntegerCoefficients(pkfac, Bi);
122        //System.out.println("aq = " + aq);
123        //System.out.println("bq = " + bq);
124
125        // compute error:
126        GenPolynomial<BigInteger> E = Ci; // - sum_i s_i b_i
127        E = E.subtract(Bi.multiply(supi.get(0)));
128        E = E.subtract(Ai.multiply(supi.get(1)));
129        //System.out.println("E     = " + E);
130        if (E.isZERO()) {
131            logger.info("liftDiophant leaving on zero E");
132            return sup;
133        }
134        GenPolynomial<MOD> Ep = PolyUtil.<MOD> fromIntegerCoefficients(pkfac, E);
135        //System.out.println("Ep(0," + pkfac.nvar + ") = " + Ep);
136        logger.info("Ep(0," + pkfac.nvar + ") = " + Ep);
137        if (Ep.isZERO()) {
138            logger.info("liftDiophant leaving on zero Ep mod p^k");
139            return sup;
140        }
141        for (int e = 1; e <= d; e++) {
142            //System.out.println("\ne = " + e + " -------------------------------------- " + pkfac.nvar);
143            GenPolynomial<GenPolynomial<MOD>> Epr = PolyUtil.<MOD> recursive(qrfac, Ep);
144            //System.out.println("Epr   = " + Epr);
145            UnivPowerSeriesRing<GenPolynomial<MOD>> psfac = new UnivPowerSeriesRing<GenPolynomial<MOD>>(
146                            qrfac);
147            //System.out.println("psfac = " + psfac);
148            TaylorFunction<GenPolynomial<MOD>> F = new PolynomialTaylorFunction<GenPolynomial<MOD>>(Epr);
149            //System.out.println("F     = " + F);
150            //List<GenPolynomial<MOD>> Vs = new ArrayList<GenPolynomial<MOD>>(1);
151            GenPolynomial<MOD> vq = ckfac.fromInteger(v.getSymmetricInteger().getVal());
152            //Vs.add(vq);
153            //System.out.println("Vs    = " + Vs);
154            UnivPowerSeries<GenPolynomial<MOD>> Epst = psfac.seriesOfTaylor(F, vq);
155            //System.out.println("Epst  = " + Epst);
156            GenPolynomial<MOD> cm = Epst.coefficient(e);
157            //System.out.println("cm   = " + cm + ", cm.ring   = " + cm.ring.toScript());
158
159            // recursion:
160            List<GenPolynomial<MOD>> S = HenselMultUtil.<MOD> liftDiophant(Ap, Bp, cm, Vp, d, k);
161            //System.out.println("S    = " + S);
162            if (!ckfac.coFac.equals(S.get(0).ring.coFac)) {
163                throw new IllegalArgumentException(
164                                "ckfac != pkfac: " + ckfac.coFac + " != " + S.get(0).ring.coFac);
165            }
166            if (pkfac.nvar == 2 && !HenselUtil.<MOD> isDiophantLift(Ap, Bp, S.get(1), S.get(0), cm)) {
167                //System.out.println("isDiophantLift: false");
168                throw new NoLiftingException("isDiophantLift: false");
169            }
170            mon = mon.multiply(xv); // Power.<GenPolynomial<MOD>> power(pkfac,xv,e);
171            //System.out.println("mon  = " + mon);
172            //List<GenPolynomial<MOD>> Sp = new ArrayList<GenPolynomial<MOD>>(S.size());
173            int i = 0;
174            supi = new ArrayList<GenPolynomial<BigInteger>>(su.size());
175            for (GenPolynomial<MOD> dd : S) {
176                //System.out.println("dd = " + dd);
177                GenPolynomial<MOD> de = dd.extend(pkfac, 0, 0L);
178                GenPolynomial<MOD> dm = de.multiply(mon);
179                //Sp.add(dm);
180                de = sup.get(i).sum(dm);
181                //System.out.println("dd = " + dd);
182                sup.set(i++, de);
183                GenPolynomial<BigInteger> spi = PolyUtil.integerFromModularCoefficients(ifac, dm);
184                supi.add(spi);
185            }
186            //System.out.println("Sp   = " + Sp);
187            //System.out.println("sup  = " + sup);
188            //System.out.println("supi = " + supi);
189            // compute new error
190            //E = E; // - sum_i s_i b_i
191            E = E.subtract(Bi.multiply(supi.get(0)));
192            E = E.subtract(Ai.multiply(supi.get(1)));
193            //System.out.println("E     = " + E);
194            if (E.isZERO()) {
195                logger.info("liftDiophant leaving on zero E");
196                return sup;
197            }
198            Ep = PolyUtil.<MOD> fromIntegerCoefficients(pkfac, E);
199            //System.out.println("Ep(" + e + "," + pkfac.nvar + ") = " + Ep); 
200            logger.info("Ep(" + e + "," + pkfac.nvar + ") = " + Ep);
201            if (Ep.isZERO()) {
202                logger.info("liftDiophant leaving on zero Ep mod p^k");
203                return sup;
204            }
205        }
206        //System.out.println("*** done: " + pkfac.nvar);
207        return sup;
208    }
209
210
211    /**
212     * Modular diophantine equation solution and lifting algorithm. Let p =
213     * A_i.ring.coFac.modul() and assume ggt(a,b) == 1 mod p, for a, b in A.
214     * @param A list of modular GenPolynomials, mod p^k
215     * @param C modular GenPolynomial, mod p^k
216     * @param V list of substitution values, mod p^k
217     * @param d desired approximation exponent (x_i-v_i)^d.
218     * @param k desired approximation exponent p^k.
219     * @return [s_1,..., s_n] with sum_i s_i A_i' = C mod p^k, with Ai' =
220     *         prod_{j!=i} A_j.
221     */
222    public static <MOD extends GcdRingElem<MOD> & Modular> List<GenPolynomial<MOD>> liftDiophant(
223                    List<GenPolynomial<MOD>> A, GenPolynomial<MOD> C, List<MOD> V, long d, long k)
224                    throws NoLiftingException {
225        GenPolynomialRing<MOD> pkfac = C.ring;
226        if (pkfac.nvar == 1) { // V, d ignored
227            return HenselUtil.<MOD> liftDiophant(A, C, k);
228        }
229        if (!pkfac.equals(A.get(0).ring)) {
230            throw new IllegalArgumentException("A.ring != pkfac: " + A.get(0).ring + " != " + pkfac);
231        }
232        // co-products
233        GenPolynomial<MOD> As = pkfac.getONE();
234        for (GenPolynomial<MOD> a : A) {
235            As = As.multiply(a);
236        }
237        List<GenPolynomial<MOD>> Bp = new ArrayList<GenPolynomial<MOD>>(A.size());
238        for (GenPolynomial<MOD> a : A) {
239            GenPolynomial<MOD> b = PolyUtil.<MOD> basePseudoDivide(As, a);
240            Bp.add(b);
241        }
242
243        // evaluate at v_n:
244        List<MOD> Vp = new ArrayList<MOD>(V);
245        MOD v = Vp.remove(Vp.size() - 1);
246        // (x_n - v)
247        GenPolynomial<MOD> mon = pkfac.getONE();
248        GenPolynomial<MOD> xv = pkfac.univariate(0, 1);
249        xv = xv.subtract(pkfac.fromInteger(v.getSymmetricInteger().getVal()));
250        //System.out.println("xv = " + xv);
251        // A(v), B(v), C(v)
252        ModularRingFactory<MOD> cf = (ModularRingFactory<MOD>) pkfac.coFac;
253        MOD vp = cf.fromInteger(v.getSymmetricInteger().getVal());
254        //System.out.println("v = " + v + ", vp = " + vp);
255        GenPolynomialRing<MOD> ckfac = pkfac.contract(1);
256        List<GenPolynomial<MOD>> Ap = new ArrayList<GenPolynomial<MOD>>(A.size());
257        for (GenPolynomial<MOD> a : A) {
258            GenPolynomial<MOD> ap = PolyUtil.<MOD> evaluateMain(ckfac, a, vp);
259            Ap.add(ap);
260        }
261        GenPolynomial<MOD> Cp = PolyUtil.<MOD> evaluateMain(ckfac, C, vp);
262        //System.out.println("Ap = " + Ap);
263        //System.out.println("Cp = " + Cp);
264
265        // recursion:
266        List<GenPolynomial<MOD>> su = HenselMultUtil.<MOD> liftDiophant(Ap, Cp, Vp, d, k);
267        //System.out.println("su@p^" + k + " = " + su);
268        //System.out.println("coFac = " + su.get(0).ring.coFac.toScript());
269        if (pkfac.nvar == 2 && !HenselUtil.<MOD> isDiophantLift(Ap, su, Cp)) {
270            //System.out.println("isDiophantLift: false");
271            throw new NoLiftingException("isDiophantLift: false");
272        }
273        if (!ckfac.equals(su.get(0).ring)) {
274            throw new IllegalArgumentException("qfac != ckfac: " + su.get(0).ring + " != " + ckfac);
275        }
276        GenPolynomialRing<BigInteger> ifac = new GenPolynomialRing<BigInteger>(new BigInteger(), pkfac);
277        //System.out.println("ifac = " + ifac.toScript());
278        String[] mn = new String[] { pkfac.getVars()[pkfac.nvar - 1] };
279        GenPolynomialRing<GenPolynomial<MOD>> qrfac = new GenPolynomialRing<GenPolynomial<MOD>>(ckfac, 1, mn);
280        //System.out.println("qrfac = " + qrfac);
281
282        List<GenPolynomial<MOD>> sup = new ArrayList<GenPolynomial<MOD>>(su.size());
283        List<GenPolynomial<BigInteger>> supi = new ArrayList<GenPolynomial<BigInteger>>(su.size());
284        for (GenPolynomial<MOD> s : su) {
285            GenPolynomial<MOD> sp = s.extend(pkfac, 0, 0L);
286            sup.add(sp);
287            GenPolynomial<BigInteger> spi = PolyUtil.integerFromModularCoefficients(ifac, sp);
288            supi.add(spi);
289        }
290        //System.out.println("sup  = " + sup);
291        //System.out.println("supi = " + supi);
292        //List<GenPolynomial<BigInteger>> Ai = new ArrayList<GenPolynomial<BigInteger>>(A.size());
293        //for (GenPolynomial<MOD> a : A) {
294        //    GenPolynomial<BigInteger> ai = PolyUtil.integerFromModularCoefficients(ifac, a);
295        //    Ai.add(ai);
296        //}
297        List<GenPolynomial<BigInteger>> Bi = new ArrayList<GenPolynomial<BigInteger>>(A.size());
298        for (GenPolynomial<MOD> b : Bp) {
299            GenPolynomial<BigInteger> bi = PolyUtil.integerFromModularCoefficients(ifac, b);
300            Bi.add(bi);
301        }
302        GenPolynomial<BigInteger> Ci = PolyUtil.integerFromModularCoefficients(ifac, C);
303        //System.out.println("Ai = " + Ai);
304        //System.out.println("Ci = " + Ci);
305
306        //List<GenPolynomial<MOD>> Aq = new ArrayList<GenPolynomial<MOD>>(A.size());
307        //for (GenPolynomial<BigInteger> ai : Ai) {
308        //    GenPolynomial<MOD> aq = PolyUtil.<MOD> fromIntegerCoefficients(pkfac, ai);
309        //    Aq.add(aq);
310        //}
311        //System.out.println("Aq = " + Aq);
312
313        // compute error:
314        GenPolynomial<BigInteger> E = Ci; // - sum_i s_i b_i
315        int i = 0;
316        for (GenPolynomial<BigInteger> bi : Bi) {
317            E = E.subtract(bi.multiply(supi.get(i++)));
318        }
319        //System.out.println("E     = " + E);
320        if (E.isZERO()) {
321            logger.info("liftDiophant leaving on zero E");
322            return sup;
323        }
324        GenPolynomial<MOD> Ep = PolyUtil.<MOD> fromIntegerCoefficients(pkfac, E);
325        //System.out.println("Ep(0," + pkfac.nvar + ") = " + Ep);
326        logger.info("Ep(0," + pkfac.nvar + ") = " + Ep);
327        if (Ep.isZERO()) {
328            logger.info("liftDiophant leaving on zero Ep mod p^k");
329            return sup;
330        }
331        for (int e = 1; e <= d; e++) {
332            //System.out.println("\ne = " + e + " -------------------------------------- " + pkfac.nvar);
333            GenPolynomial<GenPolynomial<MOD>> Epr = PolyUtil.<MOD> recursive(qrfac, Ep);
334            //System.out.println("Epr   = " + Epr);
335            UnivPowerSeriesRing<GenPolynomial<MOD>> psfac = new UnivPowerSeriesRing<GenPolynomial<MOD>>(
336                            qrfac);
337            //System.out.println("psfac = " + psfac);
338            TaylorFunction<GenPolynomial<MOD>> F = new PolynomialTaylorFunction<GenPolynomial<MOD>>(Epr);
339            //System.out.println("F     = " + F);
340            //List<GenPolynomial<MOD>> Vs = new ArrayList<GenPolynomial<MOD>>(1);
341            GenPolynomial<MOD> vq = ckfac.fromInteger(v.getSymmetricInteger().getVal());
342            //Vs.add(vq);
343            //System.out.println("Vs    = " + Vs);
344            UnivPowerSeries<GenPolynomial<MOD>> Epst = psfac.seriesOfTaylor(F, vq);
345            //System.out.println("Epst  = " + Epst);
346            GenPolynomial<MOD> cm = Epst.coefficient(e);
347            //System.out.println("cm   = " + cm + ", cm.ring   = " + cm.ring.toScript());
348            if (cm.isZERO()) {
349                continue;
350            }
351            // recursion:
352            List<GenPolynomial<MOD>> S = HenselMultUtil.<MOD> liftDiophant(Ap, cm, Vp, d, k);
353            //System.out.println("S    = " + S);
354            if (!ckfac.coFac.equals(S.get(0).ring.coFac)) {
355                throw new IllegalArgumentException(
356                                "ckfac != pkfac: " + ckfac.coFac + " != " + S.get(0).ring.coFac);
357            }
358            if (pkfac.nvar == 2 && !HenselUtil.<MOD> isDiophantLift(Ap, S, cm)) {
359                //System.out.println("isDiophantLift: false");
360                throw new NoLiftingException("isDiophantLift: false");
361            }
362            mon = mon.multiply(xv); // Power.<GenPolynomial<MOD>> power(pkfac,xv,e);
363            //System.out.println("mon  = " + mon);
364            //List<GenPolynomial<MOD>> Sp = new ArrayList<GenPolynomial<MOD>>(S.size());
365            i = 0;
366            supi = new ArrayList<GenPolynomial<BigInteger>>(su.size());
367            for (GenPolynomial<MOD> dd : S) {
368                //System.out.println("dd = " + dd);
369                GenPolynomial<MOD> de = dd.extend(pkfac, 0, 0L);
370                GenPolynomial<MOD> dm = de.multiply(mon);
371                //Sp.add(dm);
372                de = sup.get(i).sum(dm);
373                //System.out.println("dd = " + dd);
374                sup.set(i++, de);
375                GenPolynomial<BigInteger> spi = PolyUtil.integerFromModularCoefficients(ifac, dm);
376                supi.add(spi);
377            }
378            //System.out.println("Sp   = " + Sp);
379            //System.out.println("sup  = " + sup);
380            //System.out.println("supi = " + supi);
381            // compute new error
382            //E = E; // - sum_i s_i b_i
383            i = 0;
384            for (GenPolynomial<BigInteger> bi : Bi) {
385                E = E.subtract(bi.multiply(supi.get(i++)));
386            }
387            //System.out.println("E     = " + E);
388            if (E.isZERO()) {
389                logger.info("liftDiophant leaving on zero E");
390                return sup;
391            }
392            Ep = PolyUtil.<MOD> fromIntegerCoefficients(pkfac, E);
393            //System.out.println("Ep(" + e + "," + pkfac.nvar + ") = " + Ep); 
394            logger.info("Ep(" + e + "," + pkfac.nvar + ") = " + Ep);
395            if (Ep.isZERO()) {
396                logger.info("liftDiophant leaving on zero Ep mod p^k");
397                return sup;
398            }
399        }
400        //System.out.println("*** done: " + pkfac.nvar);
401        return sup;
402    }
403
404
405    /**
406     * Modular Hensel lifting algorithm on coefficients test. Let p =
407     * f_i.ring.coFac.modul() and assume C == prod_{0,...,n-1} f_i mod p with
408     * gcd(f_i,f_j) == 1 mod p for i != j
409     * @param C integer polynomial
410     * @param Cp GenPolynomial mod p^k
411     * @param F = [f_0,...,f_{n-1}] list of monic modular polynomials.
412     * @param L = [g_0,...,g_{n-1}] list of lifted modular polynomials.
413     * @return true if C = prod_{0,...,n-1} g_i mod p^k, else false.
414     */
415    @SuppressWarnings("unused")
416    public static <MOD extends GcdRingElem<MOD> & Modular> boolean isHenselLift(GenPolynomial<BigInteger> C,
417                    GenPolynomial<MOD> Cp, List<GenPolynomial<MOD>> F, List<GenPolynomial<MOD>> L) {
418        boolean t = true;
419        GenPolynomialRing<MOD> qfac = L.get(0).ring;
420        GenPolynomial<MOD> q = qfac.getONE();
421        for (GenPolynomial<MOD> fi : L) {
422            q = q.multiply(fi);
423        }
424        t = Cp.equals(q);
425        if (!t) {
426            System.out.println("Cp     = " + Cp);
427            System.out.println("q      = " + q);
428            System.out.println("Cp != q: " + Cp.subtract(q));
429            return t;
430        }
431        GenPolynomialRing<BigInteger> dfac = C.ring;
432        GenPolynomial<BigInteger> Ci = PolyUtil.integerFromModularCoefficients(dfac, q);
433        t = C.equals(Ci);
434        if (!t) {
435            System.out.println("C      = " + C);
436            System.out.println("Ci     = " + Ci);
437            System.out.println("C != Ci: " + C.subtract(Ci));
438            return t;
439        }
440        // test L mod id(V) == F
441        return t;
442    }
443
444
445    /**
446     * Modular Hensel lifting algorithm, monic case. Let p =
447     * A_i.ring.coFac.modul() and assume ggt(a,b) == 1 mod p, for a, b in A.
448     * @param C monic GenPolynomial with integer coefficients
449     * @param Cp GenPolynomial mod p^k
450     * @param F list of modular GenPolynomials, mod (I_v, p^k )
451     * @param V list of integer substitution values
452     * @param k desired approximation exponent p^k.
453     * @return [g'_1,..., g'_n] with prod_i g'_i = Cp mod p^k.
454     */
455    public static <MOD extends GcdRingElem<MOD> & Modular> List<GenPolynomial<MOD>> liftHenselMonic(
456                    GenPolynomial<BigInteger> C, GenPolynomial<MOD> Cp, List<GenPolynomial<MOD>> F,
457                    List<BigInteger> V, long k) throws NoLiftingException {
458        GenPolynomialRing<MOD> pkfac = Cp.ring;
459        //if (pkfac.nvar == 1) { // V ignored
460        //    return HenselUtil.<MOD> liftHenselMonic(C,F,k);
461        //}
462        long d = C.degree();
463        //System.out.println("d = " + d);
464        // prepare stack of polynomial rings and polynomials
465        List<GenPolynomialRing<MOD>> Pfac = new ArrayList<GenPolynomialRing<MOD>>();
466        List<GenPolynomial<MOD>> Ap = new ArrayList<GenPolynomial<MOD>>();
467        List<MOD> Vb = new ArrayList<MOD>();
468        MOD v = pkfac.coFac.fromInteger(V.get(0).getVal());
469        Pfac.add(pkfac);
470        Ap.add(Cp);
471        Vb.add(v);
472        GenPolynomialRing<MOD> pf = pkfac;
473        GenPolynomial<MOD> ap = Cp;
474        for (int j = pkfac.nvar; j > 2; j--) {
475            pf = pf.contract(1);
476            Pfac.add(0, pf);
477            //MOD vp = pkfac.coFac.fromInteger(V.get(j - 2).getSymmetricInteger().getVal());
478            MOD vp = pkfac.coFac.fromInteger(V.get(j - 2).getVal());
479            //System.out.println("vp     = " + vp);
480            Vb.add(1, vp);
481            ap = PolyUtil.<MOD> evaluateMain(pf, ap, vp);
482            Ap.add(0, ap);
483        }
484        //System.out.println("Pfac   = " + Pfac);
485        if (debug) {
486            logger.debug("Pfac   = " + Pfac);
487        }
488        //System.out.println("Ap     = " + Ap);
489        //System.out.println("V      = " + V);
490        //System.out.println("Vb     = " + Vb);
491        // setup bi-variate base case
492        GenPolynomialRing<MOD> pk1fac = F.get(0).ring;
493        if (!pkfac.coFac.equals(pk1fac.coFac)) {
494            throw new IllegalArgumentException("F.ring != pkfac: " + pk1fac + " != " + pkfac);
495        }
496        // TODO: adjust leading coefficients
497        pkfac = Pfac.get(0);
498        //Cp = Ap.get(0);
499        //System.out.println("pkfac  = " + pkfac.toScript());
500        //System.out.println("pk1fac = " + pk1fac.toScript());
501        GenPolynomialRing<BigInteger> i1fac = new GenPolynomialRing<BigInteger>(new BigInteger(), pk1fac);
502        //System.out.println("i1fac = " + i1fac.toScript());
503        List<GenPolynomial<BigInteger>> Bi = new ArrayList<GenPolynomial<BigInteger>>(F.size());
504        for (GenPolynomial<MOD> b : F) {
505            GenPolynomial<BigInteger> bi = PolyUtil.integerFromModularCoefficients(i1fac, b);
506            Bi.add(bi);
507        }
508        //System.out.println("Bi = " + Bi);
509        // evaluate Cp at v_n:
510        //ModularRingFactory<MOD> cf = (ModularRingFactory<MOD>) pkfac.coFac;
511        //MOD vp = cf.fromInteger(v.getSymmetricInteger().getVal());
512        //System.out.println("v = " + v + ", vp = " + vp);
513        GenPolynomialRing<MOD> ckfac; // = pkfac.contract(1);
514        //GenPolynomial<MOD> Cs = PolyUtil.<MOD> evaluateMain(ckfac, Cp, vp);
515        //System.out.println("Cp = " + Cp);
516        //System.out.println("Cs = " + Cs);
517
518        List<GenPolynomial<MOD>> U = new ArrayList<GenPolynomial<MOD>>(F.size());
519        for (GenPolynomial<MOD> b : F) {
520            GenPolynomial<MOD> bi = b.extend(pkfac, 0, 0L);
521            U.add(bi);
522        }
523        //System.out.println("U  = " + U);
524        List<GenPolynomial<MOD>> U1 = F;
525        //System.out.println("U1 = " + U1);
526
527        GenPolynomial<BigInteger> E = C.ring.getZERO();
528        List<MOD> Vh = new ArrayList<MOD>();
529
530        while (Pfac.size() > 0) { // loop through stack of polynomial rings
531            pkfac = Pfac.remove(0);
532            Cp = Ap.remove(0);
533            v = Vb.remove(0);
534            //Vh.add(0,v);
535            //System.out.println("\npkfac = " + pkfac.toScript() + " ================================== " + Vh);
536
537            // (x_n - v)
538            GenPolynomial<MOD> mon = pkfac.getONE();
539            GenPolynomial<MOD> xv = pkfac.univariate(0, 1);
540            xv = xv.subtract(pkfac.fromInteger(v.getSymmetricInteger().getVal()));
541            //System.out.println("xv = " + xv);
542
543            long deg = Cp.degree(pkfac.nvar - 1);
544            //System.out.println("deg = " + deg);
545
546            GenPolynomialRing<BigInteger> ifac = new GenPolynomialRing<BigInteger>(new BigInteger(), pkfac);
547            //System.out.println("ifac = " + ifac.toScript());
548            List<GenPolynomial<BigInteger>> Bip = new ArrayList<GenPolynomial<BigInteger>>(F.size());
549            for (GenPolynomial<BigInteger> b : Bi) {
550                GenPolynomial<BigInteger> bi = b.extend(ifac, 0, 0L);
551                Bip.add(bi);
552            }
553            Bi = Bip;
554            //System.out.println("Bi = " + Bi);
555            GenPolynomial<BigInteger> Ci = PolyUtil.integerFromModularCoefficients(ifac, Cp);
556            //System.out.println("Ci = " + Ci);
557
558            // compute error:
559            E = ifac.getONE();
560            for (GenPolynomial<BigInteger> bi : Bi) {
561                E = E.multiply(bi);
562            }
563            E = Ci.subtract(E);
564            //System.out.println("E     = " + E);
565            GenPolynomial<MOD> Ep = PolyUtil.<MOD> fromIntegerCoefficients(pkfac, E);
566            //System.out.println("Ep(0," + pkfac.nvar + ") = " + Ep);
567            logger.info("Ep(0," + deg + "," + pkfac.nvar + ") = " + Ep);
568
569            String[] mn = new String[] { pkfac.getVars()[pkfac.nvar - 1] };
570            ckfac = pkfac.contract(1);
571            GenPolynomialRing<GenPolynomial<MOD>> pkrfac = new GenPolynomialRing<GenPolynomial<MOD>>(ckfac, 1,
572                            mn);
573            //System.out.println("pkrfac = " + pkrfac.toScript());
574
575            for (int e = 1; e <= deg && !Ep.isZERO(); e++) {
576                //System.out.println("\ne = " + e + " -------------------------------------- " + pkfac.nvar);
577                GenPolynomial<GenPolynomial<MOD>> Epr = PolyUtil.<MOD> recursive(pkrfac, Ep);
578                //System.out.println("Epr   = " + Epr);
579                UnivPowerSeriesRing<GenPolynomial<MOD>> psfac = new UnivPowerSeriesRing<GenPolynomial<MOD>>(
580                                pkrfac);
581                //System.out.println("psfac = " + psfac);
582                TaylorFunction<GenPolynomial<MOD>> T = new PolynomialTaylorFunction<GenPolynomial<MOD>>(Epr);
583                //System.out.println("T     = " + T);
584                //List<GenPolynomial<MOD>> Vs = new ArrayList<GenPolynomial<MOD>>(1);
585                GenPolynomial<MOD> vq = ckfac.fromInteger(v.getSymmetricInteger().getVal());
586                //Vs.add(vq);
587                //System.out.println("Vs    = " + Vs + ", Vh = " + Vh);
588                UnivPowerSeries<GenPolynomial<MOD>> Epst = psfac.seriesOfTaylor(T, vq);
589                //System.out.println("Epst  = " + Epst);
590                logger.info("Epst(" + e + "," + deg + ", " + pkfac.nvar + ") = " + Epst);
591                GenPolynomial<MOD> cm = Epst.coefficient(e);
592                //System.out.println("cm   = " + cm);
593                if (cm.isZERO()) {
594                    continue;
595                }
596                List<GenPolynomial<MOD>> Ud = HenselMultUtil.<MOD> liftDiophant(U1, cm, Vh, d, k);
597                //System.out.println("Ud = " + Ud);
598
599                mon = mon.multiply(xv);
600                //System.out.println("mon  = " + mon);
601                //List<GenPolynomial<MOD>> Sd = new ArrayList<GenPolynomial<MOD>>(Ud.size());
602                int i = 0;
603                List<GenPolynomial<BigInteger>> Si = new ArrayList<GenPolynomial<BigInteger>>(Ud.size());
604                for (GenPolynomial<MOD> dd : Ud) {
605                    //System.out.println("dd = " + dd);
606                    GenPolynomial<MOD> de = dd.extend(pkfac, 0, 0L);
607                    GenPolynomial<MOD> dm = de.multiply(mon);
608                    //Sd.add(dm);
609                    de = U.get(i).sum(dm);
610                    //System.out.println("de = " + de);
611                    U.set(i++, de);
612                    GenPolynomial<BigInteger> si = PolyUtil.integerFromModularCoefficients(ifac, de);
613                    Si.add(si);
614                }
615                //System.out.println("Sd   = " + Sd);
616                //System.out.println("U    = " + U);
617                //System.out.println("Si   = " + Si);
618
619                // compute new error:
620                E = ifac.getONE();
621                for (GenPolynomial<BigInteger> bi : Si) {
622                    E = E.multiply(bi);
623                }
624                E = Ci.subtract(E);
625                //System.out.println("E     = " + E);
626                Ep = PolyUtil.<MOD> fromIntegerCoefficients(pkfac, E);
627                //System.out.println("Ep(0," + pkfac.nvar + ") = " + Ep);
628                logger.info("Ep(" + e + "," + deg + "," + pkfac.nvar + ") = " + Ep);
629            }
630            Vh.add(v);
631            U1 = U;
632            if (Pfac.size() > 0) {
633                List<GenPolynomial<MOD>> U2 = new ArrayList<GenPolynomial<MOD>>(U.size());
634                pkfac = Pfac.get(0);
635                for (GenPolynomial<MOD> b : U) {
636                    GenPolynomial<MOD> bi = b.extend(pkfac, 0, 0L);
637                    U2.add(bi);
638                }
639                U = U2;
640                //System.out.println("U  = " + U);
641            }
642        }
643        if (E.isZERO()) {
644            logger.info("liftHensel leaving with zero E");
645        }
646        return U;
647    }
648
649
650    /**
651     * Modular Hensel lifting algorithm. Let p = A_i.ring.coFac.modul() and
652     * assume ggt(a,b) == 1 mod p, for a, b in A.
653     * @param C GenPolynomial with integer coefficients
654     * @param Cp GenPolynomial C mod p^k
655     * @param F list of modular GenPolynomials, mod (I_v, p^k )
656     * @param V list of integral substitution values
657     * @param k desired approximation exponent p^k.
658     * @param G list of leading coefficients of the factors of C.
659     * @return [g'_1,..., g'_n] with prod_i g'_i = Cp mod p^k.
660     */
661    public static <MOD extends GcdRingElem<MOD> & Modular> List<GenPolynomial<MOD>> liftHensel(
662                    GenPolynomial<BigInteger> C, GenPolynomial<MOD> Cp, List<GenPolynomial<MOD>> F,
663                    List<BigInteger> V, long k, List<GenPolynomial<BigInteger>> G) throws NoLiftingException {
664        GenPolynomialRing<MOD> pkfac = Cp.ring;
665        long d = C.degree();
666        //System.out.println("C = " + C);
667        //System.out.println("Cp = " + Cp);
668        //System.out.println("G = " + G);
669
670        //GenPolynomial<BigInteger> cd = G.get(0); // 1
671        //System.out.println("cd = " + cd + ", ring = " + C.ring);
672        //if ( cd.equals(C.ring.univariate(0)) ) {
673        //    System.out.println("cd == G[1]");
674        //}
675        // G mod p^k, in all variables
676        GenPolynomialRing<MOD> pkfac1 = new GenPolynomialRing<MOD>(pkfac.coFac, G.get(0).ring);
677        List<GenPolynomial<MOD>> Lp = new ArrayList<GenPolynomial<MOD>>(G.size());
678        for (GenPolynomial<BigInteger> cd1 : G) {
679            GenPolynomial<MOD> cdq = PolyUtil.<MOD> fromIntegerCoefficients(pkfac1, cd1);
680            cdq = cdq.extendLower(pkfac, 0, 0L); // reintroduce lower variable
681            Lp.add(cdq);
682        }
683        logger.info("G modulo p^k: " + Lp); // + ", ring = " + pkfac1);
684
685        // prepare stack of polynomial rings, polynomials and evaluated leading coefficients
686        List<GenPolynomialRing<MOD>> Pfac = new ArrayList<GenPolynomialRing<MOD>>();
687        List<GenPolynomial<MOD>> Ap = new ArrayList<GenPolynomial<MOD>>();
688        List<List<GenPolynomial<MOD>>> Gp = new ArrayList<List<GenPolynomial<MOD>>>();
689        List<MOD> Vb = new ArrayList<MOD>();
690        //MOD v = V.get(0); // fromInteger
691        Pfac.add(pkfac);
692        Ap.add(Cp);
693        Gp.add(Lp);
694        GenPolynomialRing<MOD> pf = pkfac;
695        //GenPolynomialRing<MOD> pf1 = pkfac1;
696        GenPolynomial<MOD> ap = Cp;
697        List<GenPolynomial<MOD>> Lpp = Lp;
698        for (int j = pkfac.nvar; j > 2; j--) {
699            pf = pf.contract(1);
700            Pfac.add(0, pf);
701            //MOD vp = pkfac.coFac.fromInteger(V.get(pkfac.nvar - j).getSymmetricInteger().getVal());
702            MOD vp = pkfac.coFac.fromInteger(V.get(pkfac.nvar - j).getVal());
703            //System.out.println("vp     = " + vp);
704            Vb.add(vp);
705            ap = PolyUtil.<MOD> evaluateMain(pf, ap, vp);
706            Ap.add(0, ap);
707            List<GenPolynomial<MOD>> Lps = new ArrayList<GenPolynomial<MOD>>(Lpp.size());
708            for (GenPolynomial<MOD> qp : Lpp) {
709                GenPolynomial<MOD> qpe = PolyUtil.<MOD> evaluateMain(pf, qp, vp);
710                Lps.add(qpe);
711            }
712            //System.out.println("Lps = " + Lps);
713            Lpp = Lps;
714            Gp.add(0, Lpp);
715        }
716        Vb.add(pkfac.coFac.fromInteger(V.get(pkfac.nvar - 2).getVal()));
717        //System.out.println("Pfac   = " + Pfac);
718        if (debug) {
719            logger.debug("Pfac   = " + Pfac);
720        }
721        //System.out.println("Ap     = " + Ap);
722        //System.out.println("Gp     = " + Gp);
723        //System.out.println("Gp[0]  = " + Gp.get(0) + ", Gp[0].ring = " + Gp.get(0).get(0).ring);
724        //System.out.println("V      = " + V);
725        //System.out.println("Vb     = " + Vb + ", V == Vb: " + V.equals(Vb));
726
727        // check bi-variate base case
728        GenPolynomialRing<MOD> pk1fac = F.get(0).ring;
729        if (!pkfac.coFac.equals(pk1fac.coFac)) {
730            throw new IllegalArgumentException("F.ring != pkfac: " + pk1fac + " != " + pkfac);
731        }
732
733        // init recursion
734        List<GenPolynomial<MOD>> U = F;
735        //logger.info("to lift U = " + U); // + ", U1.ring = " + U1.get(0).ring);
736        GenPolynomial<BigInteger> E = C.ring.getZERO();
737        List<MOD> Vh = new ArrayList<MOD>();
738        List<GenPolynomial<BigInteger>> Si; // = new ArrayList<GenPolynomial<BigInteger>>(F.size());
739        MOD v = null;
740
741        while (Pfac.size() > 0) { // loop through stack of polynomial rings
742            pkfac = Pfac.remove(0);
743            Cp = Ap.remove(0);
744            Lpp = Gp.remove(0);
745            v = Vb.remove(Vb.size() - 1); // last in stack
746            //System.out.println("\npkfac = " + pkfac.toScript() + " ================================== " + v);
747            logger.info("stack loop: pkfac = " + pkfac.toScript() + " v = " + v);
748
749            List<GenPolynomial<MOD>> U1 = U;
750            logger.info("to lift U1 = " + U1); // + ", U1.ring = " + U1.get(0).ring);
751            U = new ArrayList<GenPolynomial<MOD>>(U1.size());
752
753            // update U, replace leading coefficient if required
754            int j = 0;
755            for (GenPolynomial<MOD> b : U1) {
756                //System.out.println("b = " + b + ", b.ring = " + b.ring);
757                GenPolynomial<MOD> bi = b.extend(pkfac, 0, 0L);
758                GenPolynomial<MOD> li = Lpp.get(j);
759                if (!li.isONE()) {
760                    //System.out.println("li = " + li + ", li.ring = " + li.ring);
761                    //System.out.println("bi = " + bi);
762                    GenPolynomialRing<GenPolynomial<MOD>> pkrfac = pkfac.recursive(pkfac.nvar - 1);
763                    //System.out.println("pkrfac = " + pkrfac);
764                    GenPolynomial<GenPolynomial<MOD>> br = PolyUtil.<MOD> recursive(pkrfac, bi);
765                    //System.out.println("br = " + br);
766                    GenPolynomial<GenPolynomial<MOD>> bs = PolyUtil.<MOD> switchVariables(br);
767                    //System.out.println("bs = " + bs + ", bs.ring = " + bs.ring);
768
769                    GenPolynomial<GenPolynomial<MOD>> lr = PolyUtil.<MOD> recursive(pkrfac, li);
770                    //System.out.println("lr = " + lr);
771                    GenPolynomial<GenPolynomial<MOD>> ls = PolyUtil.<MOD> switchVariables(lr);
772                    //System.out.println("ls = " + ls + ", ls.ring = " + ls.ring);
773                    if (!ls.isConstant() && !ls.isZERO()) {
774                        throw new RuntimeException("ls not constant " + ls + ", li = " + li);
775                    }
776                    bs.doPutToMap(bs.leadingExpVector(), ls.leadingBaseCoefficient());
777                    //System.out.println("bs = " + bs + ", bs.ring = " + bs.ring);
778                    br = PolyUtil.<MOD> switchVariables(bs);
779                    //System.out.println("br = " + br);
780                    bi = PolyUtil.<MOD> distribute(pkfac, br);
781                    //System.out.println("bi = " + bi);
782                }
783                U.add(bi);
784                j++;
785            }
786            logger.info("U with leading coefficient replaced = " + U); // + ", U.ring = " + U.get(0).ring);
787
788            // (x_n - v)
789            GenPolynomial<MOD> mon = pkfac.getONE();
790            GenPolynomial<MOD> xv = pkfac.univariate(0, 1);
791            xv = xv.subtract(pkfac.fromInteger(v.getSymmetricInteger().getVal()));
792            //System.out.println("xv = " + xv);
793
794            long deg = Cp.degree(pkfac.nvar - 1);
795            //System.out.println("deg = " + deg + ", degv = " + Cp.degreeVector());
796
797            // convert to integer polynomials
798            GenPolynomialRing<BigInteger> ifac = new GenPolynomialRing<BigInteger>(new BigInteger(), pkfac);
799            //System.out.println("ifac = " + ifac.toScript());
800            List<GenPolynomial<BigInteger>> Bi = PolyUtil.integerFromModularCoefficients(ifac, U);
801            //System.out.println("Bi = " + Bi);
802            GenPolynomial<BigInteger> Ci = PolyUtil.integerFromModularCoefficients(ifac, Cp);
803            //System.out.println("Ci = " + Ci);
804
805            // compute error:
806            E = ifac.getONE();
807            for (GenPolynomial<BigInteger> bi : Bi) {
808                E = E.multiply(bi);
809            }
810            //System.out.println("E  = " + E);
811            E = Ci.subtract(E);
812            //System.out.println("E  = " + E);
813            GenPolynomial<MOD> Ep = PolyUtil.<MOD> fromIntegerCoefficients(pkfac, E);
814            logger.info("Ep(0," + deg + "," + pkfac.nvar + ") = " + Ep);
815
816            GenPolynomialRing<GenPolynomial<MOD>> pkrfac = pkfac.recursive(1);
817            GenPolynomialRing<MOD> ckfac = (GenPolynomialRing<MOD>) pkrfac.coFac;
818            //System.out.println("pkrfac = " + pkrfac.toScript());
819
820            for (int e = 1; e <= deg && !Ep.isZERO(); e++) {
821                //System.out.println("\ne = " + e + " -------------------------------------- " + deg);
822                logger.info("approximation loop: e = " + e + " of deg = " + deg);
823                GenPolynomial<GenPolynomial<MOD>> Epr = PolyUtil.<MOD> recursive(pkrfac, Ep);
824                //System.out.println("Epr   = " + Epr);
825                UnivPowerSeriesRing<GenPolynomial<MOD>> psfac = new UnivPowerSeriesRing<GenPolynomial<MOD>>(
826                                pkrfac);
827                //System.out.println("psfac = " + psfac);
828                TaylorFunction<GenPolynomial<MOD>> T = new PolynomialTaylorFunction<GenPolynomial<MOD>>(Epr);
829                //System.out.println("T     = " + T);
830                GenPolynomial<MOD> vq = ckfac.fromInteger(v.getSymmetricInteger().getVal());
831                //System.out.println("vq    = " + vq + ", Vh = " + Vh);
832                UnivPowerSeries<GenPolynomial<MOD>> Epst = psfac.seriesOfTaylor(T, vq);
833                //System.out.println("Epst  = " + Epst);
834                logger.info("Epst(" + e + "," + deg + "," + pkfac.nvar + ") = " + Epst);
835                GenPolynomial<MOD> cm = Epst.coefficient(e);
836                if (cm.isZERO()) {
837                    //System.out.println("cm   = " + cm);
838                    continue;
839                }
840                List<GenPolynomial<MOD>> Ud = HenselMultUtil.<MOD> liftDiophant(U1, cm, Vh, d, k);
841                //System.out.println("Ud = " + Ud);
842
843                mon = mon.multiply(xv);
844                //System.out.println("mon  = " + mon);
845                //List<GenPolynomial<MOD>> Sd = new ArrayList<GenPolynomial<MOD>>(Ud.size());
846                int i = 0;
847                Si = new ArrayList<GenPolynomial<BigInteger>>(Ud.size());
848                for (GenPolynomial<MOD> dd : Ud) {
849                    //System.out.println("dd = " + dd);
850                    GenPolynomial<MOD> de = dd.extend(pkfac, 0, 0L);
851                    GenPolynomial<MOD> dm = de.multiply(mon);
852                    //Sd.add(dm);
853                    de = U.get(i).sum(dm);
854                    //System.out.println("de = " + de);
855                    U.set(i++, de);
856                    GenPolynomial<BigInteger> si = PolyUtil.integerFromModularCoefficients(ifac, de);
857                    Si.add(si);
858                }
859                //System.out.println("Sd   = " + Sd);
860                //System.out.println("U    = " + U + ", U.ring = " + U.get(0).ring);
861                //System.out.println("Si   = " + Si);
862
863                // compute new error:
864                E = ifac.getONE();
865                for (GenPolynomial<BigInteger> bi : Si) {
866                    E = E.multiply(bi);
867                }
868                E = Ci.subtract(E);
869                //System.out.println("E = " + E);
870                Ep = PolyUtil.<MOD> fromIntegerCoefficients(pkfac, E);
871                //System.out.println("Ep(0," + pkfac.nvar + ") = " + Ep);
872                logger.info("Ep(" + e + "," + deg + "," + pkfac.nvar + ") = " + Ep);
873            }
874            Vh.add(v);
875            GenPolynomial<MOD> Uf = U.get(0).ring.getONE();
876            for (GenPolynomial<MOD> Upp : U) {
877                Uf = Uf.multiply(Upp);
878            }
879            if (false && !Cp.leadingExpVector().equals(Uf.leadingExpVector())) { // not meanigfull test
880                System.out.println("\nU    = " + U);
881                System.out.println("Cp   = " + Cp);
882                System.out.println("Uf   = " + Uf);
883                //System.out.println("Cp.ring = " + Cp.ring.toScript() + ", Uf.ring = " + Uf.ring.toScript() + "\n");
884                System.out.println("");
885                //throw new NoLiftingException("no factorization, Cp != Uf");
886            }
887        }
888        if (E.isZERO()) {
889            logger.info("liftHensel leaving with zero E, Ep");
890        }
891        if (false && debug) {
892            // remove normalization required ??
893            GreatestCommonDivisorAbstract<BigInteger> ufd = GCDFactory.getImplementation(new BigInteger());
894            List<GenPolynomial<BigInteger>> Fii = new ArrayList<GenPolynomial<BigInteger>>(U.size());
895            for (GenPolynomial<BigInteger> bi : Si) {
896                GenPolynomial<BigInteger> ci = ufd.content(bi); //ufd.primitivePart(bi); // ??
897                if (!ci.isONE()) {
898                    System.out.println("bi = " + bi + ", cont(bi) = " + ci);
899                }
900                //Fii.add(ci);
901            }
902            //Si = Fii;
903            //System.out.println("Si  = " + Si);
904        }
905        logger.info("multivariate lift: U = " + U + ", of " + F);
906        return U;
907    }
908
909
910    /**
911     * Modular Hensel full lifting algorithm. Let p = A_i.ring.coFac.modul() and
912     * assume ggt(a,b) == 1 mod p, for a, b in A.
913     * @param C GenPolynomial with integer coefficients
914     * @param F list of modular GenPolynomials, mod (I_v, p )
915     * @param V list of integer substitution values
916     * @param k desired approximation exponent p^k.
917     * @param G = [g_1,...,g_n] list of factors of leading coefficients.
918     * @return [c_1,..., c_n] with prod_i c_i = C mod p^k.
919     */
920    @SuppressWarnings("unchecked")
921    public static <MOD extends GcdRingElem<MOD> & Modular> List<GenPolynomial<MOD>> liftHenselFull(
922                    GenPolynomial<BigInteger> C, List<GenPolynomial<MOD>> F, List<BigInteger> V, long k,
923                    List<GenPolynomial<BigInteger>> G) throws NoLiftingException {
924        if (F == null || F.size() == 0) {
925            return new ArrayList<GenPolynomial<MOD>>();
926        }
927        GenPolynomialRing<MOD> pkfac = F.get(0).ring;
928        //long d = C.degree();
929        // setup q = p^k
930        RingFactory<MOD> cfac = pkfac.coFac;
931        ModularRingFactory<MOD> pcfac = (ModularRingFactory<MOD>) cfac;
932        //System.out.println("pcfac = " + pcfac);
933        BigInteger p = pcfac.getIntegerModul();
934        BigInteger q = p.power(k);
935        ModularRingFactory<MOD> mcfac;
936        if (ModLongRing.MAX_LONG.compareTo(q.getVal()) > 0) {
937            mcfac = (ModularRingFactory) new ModLongRing(q.getVal());
938        } else {
939            mcfac = (ModularRingFactory) new ModIntegerRing(q.getVal());
940        }
941        //System.out.println("mcfac = " + mcfac);
942
943        // convert C from Z[...] to Z_q[...]
944        GenPolynomialRing<MOD> qcfac = new GenPolynomialRing<MOD>(mcfac, C.ring);
945        GenPolynomial<MOD> Cq = PolyUtil.<MOD> fromIntegerCoefficients(qcfac, C);
946        //System.out.println("C  = " + C);
947        //System.out.println("Cq = " + Cq);
948
949        // convert g_i from Z[...] to Z_q[...]
950        GenPolynomialRing<MOD> gcfac = new GenPolynomialRing<MOD>(mcfac, G.get(0).ring);
951        List<GenPolynomial<MOD>> GQ = new ArrayList<GenPolynomial<MOD>>();
952        boolean allOnes = true;
953        for (GenPolynomial<BigInteger> g : G) {
954            if (!g.isONE()) {
955                allOnes = false;
956            }
957            GenPolynomial<MOD> gq = PolyUtil.<MOD> fromIntegerCoefficients(gcfac, g);
958            GQ.add(gq);
959        }
960        //System.out.println("G  = " + G);
961        //System.out.println("GQ = " + GQ);
962
963        // evaluate C to Z_q[x]
964        GenPolynomialRing<MOD> pf = qcfac;
965        GenPolynomial<MOD> ap = Cq;
966        for (int j = C.ring.nvar; j > 1; j--) {
967            pf = pf.contract(1);
968            //MOD vp = mcfac.fromInteger(V.get(C.ring.nvar - j).getSymmetricInteger().getVal());
969            MOD vp = mcfac.fromInteger(V.get(C.ring.nvar - j).getVal());
970            //System.out.println("vp     = " + vp);
971            ap = PolyUtil.<MOD> evaluateMain(pf, ap, vp);
972            //System.out.println("ap     = " + ap);
973        }
974        GenPolynomial<MOD> Cq1 = ap;
975        //System.out.println("Cq1 = " + Cq1);
976        if (Cq1.isZERO()) {
977            throw new NoLiftingException("C mod (I, p^k) == 0: " + C);
978        }
979        GenPolynomialRing<BigInteger> ifac = new GenPolynomialRing<BigInteger>(new BigInteger(), pf);
980        GenPolynomial<BigInteger> Ci = PolyUtil.integerFromModularCoefficients(ifac, Cq1);
981        //System.out.println("Ci  = " + Ci);
982        GreatestCommonDivisorAbstract<BigInteger> ufd = GCDFactory.getImplementation(new BigInteger());
983        Ci = Ci.abs();
984        BigInteger cCi = ufd.baseContent(Ci);
985        Ci = Ci.divide(cCi);
986        //System.out.println("cCi = " + cCi);
987        //System.out.println("Ci  = " + Ci);
988        ////System.out.println("F.fac = " + F.get(0).ring);
989
990        // evaluate G to Z_q
991        //List<GenPolynomial<MOD>> GP = new ArrayList<GenPolynomial<MOD>>();
992        for (GenPolynomial<MOD> gq : GQ) {
993            GenPolynomialRing<MOD> gf = gcfac;
994            GenPolynomial<MOD> gp = gq;
995            for (int j = gcfac.nvar; j > 1; j--) {
996                gf = gf.contract(1);
997                //MOD vp = mcfac.fromInteger(V.get(gcfac.nvar - j).getSymmetricInteger().getVal());
998                MOD vp = mcfac.fromInteger(V.get(gcfac.nvar - j).getVal());
999                //System.out.println("vp     = " + vp);
1000                gp = PolyUtil.<MOD> evaluateMain(gf, gp, vp);
1001                //System.out.println("gp     = " + gp);
1002            }
1003            //GP.add(gp);
1004        }
1005        //System.out.println("GP = " + GP); // + ", GP.ring = " + GP.get(0).ring);
1006
1007        // leading coefficient for recursion base, for Cq1 and list GP 
1008        BigInteger gi0 = Ci.leadingBaseCoefficient(); // gq0.getSymmetricInteger();
1009        //System.out.println("gi0 = " + gi0);
1010
1011        // lift F to Z_{p^k}[x]
1012        //System.out.println("Ci = " + Ci + ", F = " + F + ", k = " + k + ", p = " + F.get(0).ring + ", gi0 = " + gi0);
1013        List<GenPolynomial<MOD>> U1 = null;
1014        if (gi0.isONE()) {
1015            U1 = HenselUtil.<MOD> liftHenselMonic(Ci, F, k);
1016        } else {
1017            U1 = HenselUtil.<MOD> liftHensel(Ci, F, k, gi0); // gi0 TODO ??
1018        }
1019        logger.info("univariate lift: Ci = " + Ci + ", F = " + F + ", U1 = " + U1);
1020        //System.out.println("U1.fac = " + U1.get(0).ring);
1021
1022        // adjust leading coefficients of U1 with F
1023        List<GenPolynomial<BigInteger>> U1i = PolyUtil.<MOD> integerFromModularCoefficients(Ci.ring, U1);
1024        //System.out.println("U1i = " + U1i);
1025        boolean t = HenselUtil.isHenselLift(Ci, q, p, U1i);
1026        //System.out.println("isLift(U1) = " + t);
1027        if (!t) {
1028            //System.out.println("NoLiftingException, Ci = " + Ci + ", U1i = " + U1i);
1029            throw new NoLiftingException("Ci = " + Ci + ", U1i = " + U1i);
1030        }
1031        MOD cC = mcfac.fromInteger(cCi.getVal());
1032        List<GenPolynomial<MOD>> U1f = PolyUtil.<MOD> fromIntegerCoefficients(F.get(0).ring, U1i);
1033        //System.out.println("U1f = " + U1f);
1034        List<GenPolynomial<MOD>> U1s = new ArrayList<GenPolynomial<MOD>>(U1.size());
1035        int j = 0;
1036        int s = 0;
1037        for (GenPolynomial<MOD> u : U1) {
1038            GenPolynomial<MOD> uf = U1f.get(j);
1039            GenPolynomial<MOD> f = F.get(j);
1040            GenPolynomial<BigInteger> ui = U1i.get(j);
1041            GenPolynomial<BigInteger> gi = G.get(j);
1042            if (ui.signum() != gi.signum()) {
1043                //System.out.println("ui = " + ui + ", gi = " + gi);
1044                u = u.negate();
1045                uf = uf.negate();
1046                s++;
1047            }
1048            j++;
1049            if (uf.isConstant()) {
1050                //System.out.println("u   = " + u);
1051                u = u.monic();
1052                //System.out.println("u  = " + u);
1053                u = u.multiply(cC);
1054                cC = cC.divide(cC);
1055                //System.out.println("u   = " + u);
1056            } else {
1057                MOD x = f.leadingBaseCoefficient().divide(uf.leadingBaseCoefficient());
1058                //System.out.println("x   = " + x + ", xi = " + x.getSymmetricInteger());
1059                if (!x.isONE()) {
1060                    MOD xq = mcfac.fromInteger(x.getSymmetricInteger().getVal());
1061                    //System.out.println("xq  = " + xq);
1062                    u = u.multiply(xq);
1063                    cC = cC.divide(xq);
1064                    //System.out.println("cC  = " + cC);
1065                }
1066            }
1067            U1s.add(u);
1068        }
1069        //if ( s % 2 != 0 || !cC.isONE()) {
1070        if (!cC.isONE()) {
1071            throw new NoLiftingException("s = " + s + ", Ci = " + Ci + ", U1i = " + U1i + ", cC = " + cC);
1072        }
1073        U1 = U1s;
1074        U1i = PolyUtil.<MOD> integerFromModularCoefficients(Ci.ring, U1);
1075        //System.out.println("U1i = " + U1i);
1076        U1f = PolyUtil.<MOD> fromIntegerCoefficients(F.get(0).ring, U1i);
1077        if (!F.equals(U1f)) { // evtl loop until reached
1078            System.out.println("F   = " + F);
1079            System.out.println("U1f = " + U1f);
1080            throw new NoLiftingException("F = " + F + ", U1f = " + U1f);
1081        }
1082        logger.info("multivariate lift: U1 = " + U1);
1083
1084        // lift U to Z_{p^k}[x,...]
1085        //System.out.println("C = " + C + ", U1 = " + U1 + ", V = " + V + ", k = " + k + ", q = " + U1.get(0).ring + ", G = " + G);
1086        List<GenPolynomial<MOD>> U = null;
1087        if (allOnes) {
1088            U = HenselMultUtil.<MOD> liftHenselMonic(C, Cq, U1, V, k);
1089        } else {
1090            U = HenselMultUtil.<MOD> liftHensel(C, Cq, U1, V, k, G);
1091        }
1092        logger.info("multivariate lift: C = " + C + ", U1 = " + U1 + ", U = " + U);
1093        //System.out.println("U  = " + U);
1094        //System.out.println("U.fac = " + U.get(0).ring);
1095        return U;
1096    }
1097
1098}