very slow multiplication/division in rational function field

[seelisch on behalf of Albert Heinle]

The following takes very long in SINGULAR 3-1-2:

ring R = (0,q),(theta),dp;
poly entry = theta17+(-q15-2*q14-3*q13-4*q12-5*q11-6*q10-7*q9-8*q8-9*q7-10*q6-11*q5-12*q4-13*q3-14*q2-15*q-16)*theta16+(q29+3*q28+7*q27+13*q26+22*q25+34*q24+50*q23+70*q22+95*q21+125*q20+161*q19+203*q18+252*q17+308*q16+372*q15+428*q14+476*q13+515*q12+545*q11+565*q10+575*q9+574*q8+562*q7+538*q6+502*q5+453*q4+391*q3+315*q2+225*q+120)*theta15+(-q42-4*q41-11*q40-25*q39-50*q38-91*q37-155*q36-250*q35-386*q34-575*q33-831*q32-1170*q31-1611*q30-2175*q29-2871*q28-3710*q27-4690*q26-5810*q25-7056*q24-8414*q23-9856*q22-11354*q21-12865*q20-14345*q19-15735*q18-16974*q17-17985*q16-18689*q15-18990*q14-18909*q13-18465*q12-17689*q11-16610*q10-15270*q9-13709*q8-11980*q7-10135*q6-8239*q5-6356*q4-4564*q3-2940*q2-1575*q-560)*theta14+(q54+5*q53+16*q52+41*q51+92*q50+187*q49+353*q48+628*q47+1065*q46+1735*q45+2732*q44+4177*q43+6225*q42+9056*q41+12882*q40+17939*q39+24481*q38+32771*q37+43075*q36+55639*q35+70682*q34+88372*q33+108804*q32+131974*q31+157756*q30+185860*q29+215806*q28+246985*q27+278631*q26+309893*q25+339819*q24+367427*q23+391701*q22+411675*q21+426428*q20+435181*q19+437306*q18+432423*q17+420422*q16+401575*q15+376559*q14+346582*q13+312861*q12+276645*q11+239149*q10+201578*q9+165048*q8+130611*q7+99177*q6+71526*q5+48230*q4+29666*q3+15925*q2+6825*q+1820)*theta13+(-q65-6*q64-22*q63-63*q62-155*q61-343*q60-701*q59-1345*q58-2451*q57-4278*q56-7198*q55-11733*q54-18589*q53-28699*q52-43265*q51-63799*q50-92152*q49-130543*q48-181565*q47-248179*q46-333673*q45-441596*q44-575643*q43-739501*q42-936619*q41-1170014*q40-1441997*q39-1753919*q38-2105911*q37-2496652*q36-2923127*q35-3380499*q34-3861979*q33-4358819*q32-4860402*q31-5354477*q30-5827501*q29-6265239*q28-6653507*q27-6978743*q26-7228746*q25-7393190*q24-7464241*q23-7436963*q22-7309763*q21-7084549*q20-6766941*q19-6366119*q18-5894636*q17-5367870*q16-4803365*q15-4219710*q14-3635296*q13-3066522*q12-2527597*q11-2030222*q10-1583491*q9-1193660*q8-864214*q7-595790*q6-386386*q5-231504*q4-124579*q3-57330*q2-20475*q-4368)*theta12+(q75+7*q74+29*q73+92*q72+247*q71+590*q70+1292*q69+2643*q68+5116*q67+9457*q66+16810*q65+28874*q64+48105*q63+77963*q62+123209*q61+190245*q60+287494*q59+425804*q58+618866*q57+883609*q56+1240541*q55+1713978*q54+2332107*q53+3126867*q52+4133571*q51+5390227*q50+6936520*q49+8812464*q48+11056669*q47+13704280*q46+16784594*q45+20318470*q44+24315607*q43+28771888*q42+33666946*q41+38962309*q40+44600023*q39+50502149*q38+56571139*q37+62691244*q36+68730935*q35+74546533*q34+79986722*q33+84898120*q32+89131550*q31+92548805*q30+95029441*q29+96477351*q28+96826185*q27+96043174*q26+94131647*q25+91131660*q24+87119171*q23+82203392*q22+76522667*q21+70238707*q20+63529796*q19+56582713*q18+49584249*q17+42712395*q16+36128059*q15+29967652*q14+24337808*q13+19312634*q12+14935096*q11+11218988*q10+8151858*q9+5698407*q8+3804801*q7+2403115*q6+1416415*q5+763763*q4+365365*q3+147147*q2+45045*q+8008)*theta11+(-q84-8*q83-37*q82-129*q81-376*q80-966*q79-2258*q78-4902*q77-10025*q76-19511*q75-36402*q74-65457*q73-113910*q72-192472*q71-316620*q70-508215*q69-797473*q68-1225306*q67-1846017*q66-2730299*q65-3968427*q64-5673536*q63-7984777*q62-11070116*q61-15128465*q60-20390819*q59-27120017*q58-35608761*q57-46175479*q56-59157726*q55-74902862*q54-93755945*q53-116044932*q52-142063382*q51-172051120*q50-206173563*q49-244500607*q48-286986083*q47-333449096*q46-383558571*q45-436822461*q44-492582939*q43-550018832*q42-608156141*q41-665887167*q40-721997807*q39-775202852*q38-824187996*q37-867657138*q36-904382945*q35-933258541*q34-953347368*q33-963928871*q32-964536834*q31-954987850*q30-935397536*q29-906183073*q28-868050862*q27-821970338*q26-769135827*q25-710918057*q24-648808233*q23-584357073*q22-519112032*q21-454555728*q20-392049009*q19-332781273*q18-277731388*q17-227641117*q16-183002853*q15-144062127*q14-110834647*q13-83135184*q12-60614741*q11-42799119*q10-29127956*q9-18992545*q8-11771474*q7-6862141*q6-3707704*q5-1817816*q4-782782*q3-280280*q2-75075*q-11440)*theta10+(q92+9*q91+46*q90+175*q89+551*q88+1517*q87+3775*q86+8677*q85+18703*q84+38212*q83+74581*q82+139877*q81+253243*q80+444211*q79+757189*q78+1257386*q77+2038445*q76+3232029*q75+5019547*q74+7646140*q73+11436924*q72+16815312*q71+24323016*q70+34641066*q69+48610873*q68+67254061*q67+91789450*q66+123645277*q65+164464508*q64+216101017*q63+280604281*q62+360190441*q61+457197865*q60+574025922*q59+713056365*q58+876557703*q57+1066573992*q56+1284800887*q55+1532453056*q54+1810128450*q53+2117675954*q52+2454073820*q51+2817326941*q50+3204391243*q49+3611133043*q48+4032330356*q47+4461721846*q46+4892107024*q45+5315498836*q44+5723326840*q43+6106686025*q42+6456623000*q41+6764448368*q40+7022061523*q39+7222273135*q38+7359108966*q37+7428079054*q36+7426396980*q35+7353135921*q34+7209310506*q33+6997877688*q32+6723653256*q31+6393146079*q30+6014316858*q29+5596273275*q28+5148917160*q27+4682563401*q26+4207550601*q25+3733862769*q24+3270780702*q23+2826579105*q22+2408283186*q21+2021494926*q20+1670295297*q19+1357224099*q18+1083335088*q17+848318658*q16+650680536*q15+487961232*q14+356978589*q13+254074425*q12+175348338*q11+116865243*q10+74831757*q9+45737406*q8+26458575*q7+14324310*q6+7145853*q5+3212352*q4+1257828*q3+405405*q2+96525*q+12870)*theta9+(-q99-10*q98-56*q97-231*q96-782*q95-2299*q94-6074*q93-14751*q92-33445*q91-71586*q90-145843*q89-284605*q88-534639*q87-970707*q86-1709091*q85-2926134*q84-4883022*q83-7958118*q82-12688161*q81-19819543*q80-30370620*q79-45705588*q78-67619803*q77-98435565*q76-141106293*q75-199325748*q74-277637550*q73-381538692*q72-517569184*q71-693378528*q70-917758545*q69-1200631327*q68-1552980935*q67-1986718008*q66-2514467898*q65-3149275373*q64-3904222333*q63-4791959202*q62-5824155954*q61-7010884530*q60-8359950639*q59-9876199091*q58-11560822647*q57-13410709233*q56-15417866033*q55-17568960466*q54-19845017463*q53-22221309147*q52-24667467342*q51-27147841107*q50-29622110916*q49-32046158421*q48-34373176970*q47-36554993654*q46-38543559373*q45-40292550458*q44-41759014716*q43-42904987186*q42-43698997388*q41-44117391022*q40-44145394986*q39-43777865188*q38-43019670206*q37-41885681598*q36-40400361216*q35-38596956954*q34-36516339240*q33-34205530818*q32-31715999202*q31-29101795368*q30-26417630514*q29-23716985664*q28-21050345182*q27-18463636290*q26-15996940992*q25-13683529134*q24-11549242308*q23-9612238032*q22-7883084208*q21-6365175366*q20-5055425826*q19-3945181834*q18-3021285630*q17-2267219328*q16-1664257320*q15-1192561524*q14-832164102*q13-563796684*q12-369542514*q11-233304720*q10-141098386*q9-81180528*q8-44038137*q7-22259094*q6-10314447*q5-4281420*q4-1537107*q3-450450*q2-96525*q-11440)*theta8+(q105+11*q104+67*q103+298*q102+1080*q101+3379*q100+9453*q99+24196*q98+57569*q97+128787*q96+273231*q95+553438*q94+1075997*q93+2016735*q92+3657192*q91+6436037*q90+11019717*q89+18397189*q88+30003973*q87+47880870*q86+74872383*q85+114869036*q84+173096283*q83+256450472*q82+373879259*q81+536799939*q80+759544391*q79+1059813880*q78+1459121011*q77+1983190069*q76+2662281192*q75+3531398853*q74+4630341477*q73+6003547268*q72+7699692101*q71+9770999253*q70+12272228155*q69+15259320516*q68+18787697047*q67+22910216326*q66+27674828551*q65+33121980096*q64+39281848637*q63+46171511745*q62+53792172793*q61+62126584715*q60+71136823074*q59+80762563409*q58+90920012755*q57+101501630586*q56+112376750037*q55+123393176185*q54+134379795971*q53+145150185558*q52+155507148257*q51+165248062169*q50+174170864688*q49+182080454274*q48+188795252141*q47+194153640367*q46+198019981033*q45+200289925212*q44+200894741748*q43+199804433264*q42+197029459520*q41+192620953188*q40+186669386562*q39+179301725204*q38+170677181240*q37+160981751274*q36+150421786460*q35+139216891710*q34+127592483930*q33+115772353582*q32+103971568098*q31+92390030732*q30+81206964790*q29+70576534794*q28+60624746714*q27+51447694586*q26+43111145244*q25+35651383472*q24+29077179220*q23+23372689230*q22+18501069995*q21+14408558395*q20+11028771237*q19+8286985223*q18+6104182964*q17+4400686554*q16+3099245270*q15+2127493720*q14+1419747120*q13+918147362*q12+573212794*q11+343872738*q10+197084030*q9+107132168*q8+54719808*q7+25940200*q6+11223212*q5+4327323*q4+1434433*q3+385385*q2+75075*q+8008)*theta7+(-q110-12*q109-79*q108-377*q107-1457*q106-4836*q105-14282*q104-38408*q103-95585*q102-222755*q101-490512*q100-1027858*q99-2061348*q98-3974921*q97-7398548*q96-13335851*q95-23342939*q94-39772696*q93-66100395*q92-107347509*q91-170620609*q90-265780891*q89-406256761*q88-610006662*q87-900631638*q86-1308626820*q85-1872748088*q84-2641454830*q83-3674372459*q82-5043699859*q81-6835468267*q80-9150540598*q79-12105225506*q78-15831370284*q77-20475792894*q76-26198917672*q75-33172493074*q74-41576294268*q73-51593749027*q72-63406472134*q71-77187750422*q70-93095085649*q69-111261972846*q68-131789163843*q67-154735735031*q66-180110340164*q65-207863078032*q64-237878436148*q63-269969780853*q62-303875847919*q61-339259643405*q60-375710091709*q59-412746667621*q58-449827124838*q57-486358289884*q56-521709734617*q55-555229980825*q54-586264736452*q53-614176524327*q52-638364950461*q51-658286778382*q50-673474935703*q49-683555584035*q48-688262435939*q47-687447601958*q46-681088393321*q45-669289685002*q44-652281650056*q43-630412897910*q42-604139273828*q41-574008790372*q40-540643351434*q39-504718083850*q38-466939201686*q37-428021387818*q36-388665682772*q35-349538822179*q34-311254865006*q33-274359811613*q32-239319732987*q31-206512732148*q30-176224848533*q29-148649810169*q28-123892349306*q27-101974636229*q26-82845261439*q25-66390113996*q24-52444464455*q23-40805563557*q22-31245109466*q21-23521011382*q20-17387978784*q19-12606585145*q18-8950583389*q17-6212378502*q16-4206681655*q15-2772472065*q14-1773471986*q13-1097393517*q12-654242447*q11-373966384*q10-203716799*q9-104963287*q8-50659180*q7-22613305*q6-9176167*q5-3303300*q4-1017016*q3-252252*q2-45045*q-4368)*theta6+(q114+13*q113+92*q112+469*q111+1926*q110+6756*q109+20972*q108+58978*q107+152775*q106+369050*q105+839288*q104+1810449*q103+3726832*q102+7357527*q101+13988283*q100+25701547*q99+45773588*q98+79222619*q97+133546121*q96+219688458*q95+353280515*q94+556190572*q93+858418990*q92+1300357646*q91+1935417677*q90+2833005498*q89+40817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;
poly factor = 1 / (q^(66));
factor*entry;

[seelisch]

I set this ticket to "somebody" for the next meeting. Afterwards, it may be closed (although there are some issues which the team has to take care of in the further development in Spielwiese!).

1) I ran the above pathological example in the current SINGULAR svn trunk version on my PC (Intel i7, 2.8 GHz, 16 GB, Ubuntu 10.04) and it takes 63 sec to compute the product (factor*entry) and print it to the SINGULAR console. I gathered more detailed information inside the procedure iparith.cc:jjTIMES_P:

21 sec for pMult, 21 sec for pNormalize, 21 sec for printing the result (which calls pNormalize again).

In the svn trunk version, there does not seem to be a means to "remember" whether a polynomial over Q has already been normalized or not.

2) I re-implemented algebraic and transcendental field extensions (over arbitrary fields, i.e., allowing for extensions towers now) in the Spielwiese. There are now two implementations; one for algebraic extensions (in one variable with a min poly), and one for rational functions fields in numerous variables. (In svn trunk, there is basically just one implementation for both types with case differentiations in almost every procedure.) The new Spielwiese implementation of rational function fields has a heuristic cancellation strategy. The actual cancellation step is then done by 'factory'-calls.

3) I then ran the above example in Spielwiese and the result is now instantaneous (just run polys_test to verify this). Also, the result is the same as with the svn trunk version. (I compared using 'diff'.) Moreover, I tested with large random examples and the result was always instantaneous and looked plausible.

4) The entire Spielwiese code makes only 6 calls to p_Normalize (and only for polynomials over Q); all of them in transext.cc:handleNestedFractionsOverQ.

5) I strongly recommend to re-think the use of p_Normalize at all. (It triggers the cancellation of rational coefficients.) The goal should be to entirely hide this issue from the user of the 'coeffs' module. Up to now, this is not the case, as I need to explicitely call p_Normalize for polys over Q in order to be able to compute correct gcd's of integer coefficients of such polys. (I need this to be able to remove nested fractions in elements of Q(t_1, ..., t_n), for some n.)

[burcin]

This is probably related to #154.

[Oleksandr]

should be fixed via libpolys (Frank's rational function fields?) => postponed to Singular/libpolys

[steidel]

Is already fixed in "Spielwiese", won't be fixed in classical SINGULAR.