How do I extract the variables from a polynomial?

Questions
1 
In[1] := -3591176434993748384348496449605057311 -
931317410939396433823492399456035672 c1 -
9952543970861457213652467061541331498 c2 +
4048090288340722001114952049169242965 c3 +
397164398434703247826095886842412575 px +
1066996793775684645797090668168460265 c2 px -
460328625674890624429065259310152110 c3 px +
15946921216526762184564517273794735 py -
35440141634773840046659693982601120 c2 py +
22495307563938104656216195952976885 c3 py +
46746177228030459461263589208063870 pz -
36428604512265070024530336222388650 c3 pz -
336574847215337605742315849219352600 qx -
856298637445659071245670123987864572 c2 qx +
381082557957344248171016160868208816 c3 qx +
20336384809932402456728418013439130 qy +
79580434879454594169496332062731380 c2 qy +
46111843705049866684186439724551130 c3 qy -
754447730345013048895996548762560 qz -
54862758834533020637102302666976820 c2 qz +
109847021021547323599497109075146120 c3 qz

Out[1] = {1, c1, c2, c3, px, c2 px, c3 px, py, c2 py, c3 py, pz, c3 pz, qx, c2 qx, c3 qx, qy, c2 qy, c3 qy, qz, c2 qz, c3 qz}

My question: How to acquire the Out from the In with the help of Mathematica?

2

You can use a rule to replace all numeric parts in the expression with 1:

expr = -3591176434993748384348496449605057311 -
931317410939396433823492399456035672 c1 -
9952543970861457213652467061541331498 c2 +
4048090288340722001114952049169242965 c3 +
397164398434703247826095886842412575 px +
1066996793775684645797090668168460265 c2 px -
460328625674890624429065259310152110 c3 px +
15946921216526762184564517273794735 py -
35440141634773840046659693982601120 c2 py +
22495307563938104656216195952976885 c3 py +
46746177228030459461263589208063870 pz -
36428604512265070024530336222388650 c3 pz -
336574847215337605742315849219352600 qx -
856298637445659071245670123987864572 c2 qx +
381082557957344248171016160868208816 c3 qx +
20336384809932402456728418013439130 qy +
79580434879454594169496332062731380 c2 qy +
46111843705049866684186439724551130 c3 qy -
754447730345013048895996548762560 qz -
54862758834533020637102302666976820 c2 qz +
109847021021547323599497109075146120 c3 qz;
expr /. _?NumericQ -> 1

(Skim through the Learning Resources listed in the Rules guide to get a hang of how Rules work)

That replacement will result in:

1 + c1 + c2 + c3 + px + c2 px + c3 px + py + c2 py + c3 py + pz +
c3 pz + qx + c2 qx + c3 qx + qy + c2 qy + c3 qy + qz + c2 qz + c3 qz

Which is pretty close, only difference is that it’s “a + b + c” instead of “{a,b,c}” Applying List to the expression will fix this

List @@ (expr /. _?NumericQ -> 1)
(* {1, c1, c2, c3, px, c2 px, c3 px, py, c2 py, c3 py, pz, c3 pz, qx,
c2 qx, c3 qx, qy, c2 qy, c3 qy, qz, c2 qz, c3 qz} *)

You can look at the FullForm of the last two to see what happened, “Plus[a,b,c]” became “List[a,b,c]”

FullForm[expr /. _?NumericQ -> 1]
FullForm[List@@(expr/._?NumericQ->1)]
3

Similar to Simon Schmidt, but using MonomialList:

expr = (* long input expression from above *) ;

MonomialList[expr] /. _Integer -> 1 // Sort

This also returns:

{1, c1, c2, c3, px, c2 px, c3 px, py, c2 py, c3 py, pz, c3 pz, qx, c2 qx, c3 qx, qy, c2 qy, c3 qy, qz, c2 qz, c3 qz}
4

Thanks to help from Mr. Simon Schmidt and Mr. Arnoud Buzing.However,another problem arise about the quadratic component.
For example

expr =-706195094000394411609737806905802758216754976753979869659028490072021\
8426948359642040245441156057529091892655574765196743335253461498910119\
1637464217681227157444178174920745739217186711976920776652189897362939\
5226687200492487026481592886256327265758087279193999606847879128145367\
7947122941562523585331021859124563736502630637152851867 - 
 414452773836320077456685307580418418269900926108704086112518150973423\
4358800467908684550831600478243566185118403644236952710238377199259855\
9753960195271217546484531699572260801289374372434868101113176267807401\
4016681734708728970336659873111993345757338403660417239274562350026411\
1773406283652234482964431639579792781547485318388759088 c1 + 
 642397193462473886082092816305478178259035950111665320966407570852878\
1941125067487595673842314175673845967156704035437884699182620954246079\
7313575740367200953127292283421385677605678672506037110472819559083703\
0414997771203614302157347355385890734070963640112937345905842006466546\
888709533468725742033517008613047153218301011201334396 c1^2 - 
 331297502896188912618974447982368381249097942032320767648205008525270\
5998117688534814470567736853009020881555674855896962598803573274831918\
5204428713135015976136982957414209343179447937225852118736839432065407\
7018427143174886680005080708972203730337611305357081661039096721118462\
1332631170140148262045981002944835762282608998681818669 c2 - 
 106687544839597397900932643788153660052288090558910838709935669271829\
4150562379438989017349386708682678874268768571432149075868204168408876\
4590389741733986380023665680876710667605371081965468616482812681161424\
8297238930853292548689635026063952392477097174603276921097723049254597\
7912082879538737655579668942774101031488450710431513584 c1 c2 - 
 183853826253445734915081844765048986625508296656132321361588290062536\
6764505961760172206429157501890545706501460595671655894282454348927990\
9583030281531204450740735531953486798969447188386671168656658370397120\
7832877379927868189780898021931652876915744446164181678604000972385669\
5573806825456566757970201201371901054723494041091936296 c2^2 + 
 221737839485020949982972298194455044102553983245347505927434236890509\
1227044802677562764521014388445492574668765549072424351202979362077047\
0759208435307940850829013136200776593093272844656072633496077404526802\
2272580300188501627982811671089099764352470706506078333357150953013994\
29376993890397395145005883374431964291302413205335457368 c3 + 
 272008926154217831770479275077077575639372303047976751248891415596468\
3404433339928462695021280268258139133506796150653223731270001910365640\
0449690960928947892869473640162704518045477337401794938565943310309002\
3923429485849143798421003207419284733305122198969551971743386804637476\
010035501463332471928054906486676423455583094028972280 c1 c3 + 
 189971651708303600825518085799391291466189141991573149373016374878237\
8826532687271262721437384263708978922520920205953687450787586356357698\
7500458082807409777780713693501472157643125027095703081976838766947008\
1239785579717439931702976444738017341188079722487914996423147384440921\
591194742556939676790728929176553630176555767401859970 c2 c3 - 
 347590161632352546669613983409125025200959176870266441303278405295419\
2135424466307269268822194787465279555122610309700519427185699824906215\
7883800764984472210913024468868693275351849167333120876277362133075772\
2539265591492010718420562211864269092662874554039876474521650035267845\
0056881364442792068708552525620276984025863040353911796 c3^2 + 
 824624866225813180308867794556187561926813422633245272942927524639126\
9179718565287909772307836247961424887830262157679343195506509968645531\
4043737610481921808016425973785000323038296065033535915280989768746909\
5957973029312591762975676335430635342540222075189210665385956112761519\
982472135699317238754654146594305630324366331316309885 px - 
 240522733155577859298831660898917822780648080688007102347117364750899\
7792425783439612165062238683975228439068911033779035310625476947646903\
0776401291868885287476597173396402856795168947079371981556226339080801\
5031084420288840389298896583598063421510749500605267419875390129151065\
4227726681059264991558743723073857979786328460688701045 c3 px + 
 164930988192613243272190598213046566196407928075193655495314541299023\
1062932935844164974830759970359790318990323551867087749513456423030934\
7122541968771157180955232282355509099526945985973079162353823163759367\
9099554219865217467728567328072723037806911977616352484908281816961135\
193335948987003701802993436237413028646587860303590525 py - 
 207091153337609167229295368164252668360618479433660029690842682165582\
2326944041914976777432904204648064984993822661797687090152686534049406\
1197384122662644082386458485528021006999256482211516852695941413077071\
8859810550377564268833856880715897207716314679314385752111699812214773\
803636065449708973082206350152210159440904191416016665 c2 py - 
 244915926226544475054961071147043687531523702644837224326837217468664\
1117655827680686247185603746064778701563125431627251374109286559853824\
3808819790034262359487797573349773034975572907001523931909215700640006\
4819196295770368405693430975531058815437059305227609052361206669289734\
68243879035432782375944113293677498739244104361177930 c3 py + 
 174739390531356938493904766104162491442934056374254387332224571524397\
0661423785669557075786005481945959141233723318556000579398618412556966\
4698109825235233132696134216243623254800171491570040070562400677937582\
6992134275057445535078282453685636174306290860232870991160831948271804\
261609291068321601844511826979819444763860352548013800 pz - 
 173085255382804076090583207501963946946367360151845724281131700205507\
5222417176174508937343815184660669158779649883122609689639619840189154\
4917429634294920205384253017628699271012948108218237267012081177945746\
0096976122780414270635983006354320177609962355123055383504222503745360\
408427831668798870922246774571929554511988478399550800 c3 pz - 
 663746350073602124651057210793167909898495605869526929689722586761681\
1606856615064263263118131548657972602187601555806368318542068556049716\
4749535639415157167928338892400099001756369429231537308066682301934254\
1314402229446616614603956182410282288268862436040543534520941901326431\
545140071660583016367313739393378127930091682902287948 qx + 
 477991902040178833830198777501676671343597315228560685692984813738056\
2079540081260783288556031897337749901198089538816603761243109944768219\
2751399131243203349829161731593257847770142702442578789580638646271239\
6169398744569738514021065581467984995470459438162187943873572334831559\
95073600812804223948652188166770121437224489417473526 c2 qx + 
 216399120458987587414542137376150239670293294153065588663970597524291\
7942201911233340197192378765932486859056768260349257768900266991406037\
3209657796034146763319711503698224523959088544063048722916492656456599\
8896079766616330775232548366637511507493598106540569040147767313988068\
1670402150578401153555024999229904481213443583280754980 c3 qx + 
 173304871892676914398869667835418886204651869545884967434256491582574\
7592464090233531666330506097798325618168499949206608655609237153578748\
6083206896616252610623697161112399232303921313888192642947513604361713\
7163593051612954902697867579671966625829571164102156823217547715347176\
1048219788514699964986937595455786885183810862918270 px qx - 
 289353393814958371177356391924604285200964394805434330276297042360650\
7343628279989127772803219050896768849832220908463474993568459115942277\
1545510457483535959373854142801849151823138078476142118507462450653384\
2582549111329346477097292784092437893749615394748243568232191912609149\
0858153495444722307125877816941356000365149298326820 py qx + 
 496078373806825033939674800376818893003166977921616088664383294694844\
4279628151283093299132795916660286874186916399414535160182469417544996\
7818807381490029849723425386886942974209130041470618045585780914250745\
0434666299217515008218963780719632691956845528855842798216503551656409\
217825333654837500447225418975135024609619808921760 pz qx - 
 803943478886372443319937953454066004559372379759420000045994143759539\
2372404044464835848330545702177656281649596552863106351415637738904930\
0581030293163698601118652564850976643268344284594573154433344411305719\
8634042872856126935166930229570694749832297967955711360728951280935894\
6263577302686746885805957275462617063099186358795630 qy + 
 263611473138683063799062262726564940064006337061467440490020457746887\
6113034255808109093503697253571528002387695334792104510304513146469599\
4930793760003454818766355079864954755112479016230742150228602550187337\
8565852920991680601507134786287002206773622517786164089499132173196318\
364339190188353477923858528168370319600956982320352280 c2 qy - 
 141728498697413855110356584739239338529797052214995300470295806299789\
1831950334442371645371526724996572803827698149263948857826859613117446\
3573274069876869683036657134411374195532857094267313753091747583358639\
0012580694744638569401596016990583943515006735696232264163165989609473\
284151840516156599809492172407958046800879029140126970 c3 qy - 
 121144529004999893267437515494994671078701809642729698664310313703681\
0954117230301001783662650702780459017702726668021795299908630107220462\
3351049732388582992907124350804025426611429198489492138935441461033148\
6338489937423917900450847998894616133567091336297740499539920295889586\
4231249716816219318513669367229192787585603704821525 px qy + 
 212601387611233338816157076127589766893757264235381387259684984558794\
7338943186758799956706459679097256311706248791525963441715958240688012\
4459352280464492734440506152527625718108642811368081805458690644929008\
7228875664272974062975701352513165289219459350548121144520890964286649\
4589659830455942485458808221570884130600953052405250 py qy - 
 109112842653699450694345014068182659496201427839189924116966359056136\
7567821167857408588004068505932185732897837162497156862118569702960158\
7113510270999550588178459578600564194922741046143654128973099240412556\
7423852525559389944922955659367675150140110949594673839468171900299075\
0340903783078647210859222370878192937040804962489750 pz qy + 
 166039827474097013502077701451731661794618012109332743395695475661343\
7484795654266060639557186286857409893947493626296674372604489028731638\
2340777240882002636724817170493467074142556238577262145291343497133163\
1913849861169317971400406082285849605459618018640836426357452846498376\
7713516569640422815811844906914795402949888959578780 qx qy + 
 763724144967099619885516247196005666076025919410630811131394908705555\
5531510292478060475849272078029587875723657003096157879509103365877380\
3969178115442079157422807011464711941734333164844167987226621545122525\
1951962780882474811137529679513049861500374055091797069484083545770471\
334151921794865858503319747423219060016825741962940 qy^2 - 
 205354990910606905082127759852668020320933993294710053918579711104354\
8287371620303699568298733995628879084564347649375351623147197150534856\
4013346300871481612365925407543368198103322479754213266889147588559863\
6645876652629710089505118647071119415108832162572499748268047658482533\
140118162807575295142009494463418796817276211338688420 qz + 
 112350837973130861145392352659973574123425684627029229023261153924156\
7069691138666107846493230600200656203266552893607039281347905821460057\
7961069463658030030857912392641327828936825599767690607517909777384843\
2892665875238065222057271851818351351649363673979668491024248055697184\
389164671938472797502937531978399801360390609004909940 c2 qz + 
 281543199751220349835300104796358217518519752829842031581801215982401\
0462582710509027056229443545701175090599394812548516056239207295079288\
2605356296270345335614532250485613911328453453816515338155566374732833\
8630238245028330996802094934178086666578085173982861097428425111927274\
561485234324274511717132851857236023466696137114163600 c3 qz + 
 220682096401064877359971599366519819432397380066821706284016388891528\
2357761403552244844783327714847836568306984777948073683776462391547484\
9513399678728809410392887180130323955076392086519693899802152680967310\
6801172576267550364008919468931032637585241303182142098458273625267471\
86302004858166542979430073077181910423346230451829450 px qz - 
 789885094463641617907699224264598575159138221194843836241058102302812\
2896194542833863139911922928025046785890156991363790749406777221761313\
9703873436203274150832439786290770741175054278584912560310863941115595\
3405884718719857532827276116264628968133353858109690699853146339472822\
45904787632619490269741329116264273594556035877350 py qz + 
 504160528342973190215635757233260925668116728789866861603652600375608\
0675626460529980630916802809146845540702195608324107220239003086789597\
1429576771792563306274290495006114009762328824789966586028937399955534\
4196835022019718443768709931038123902679676332533683398612575517914749\
327605772860817795314372962541750342868356097077300 pz qz - 
 175526483812283171233178036423556053374054086239377018921440653299963\
3191757973386972114652222429490214450552002563555044743887121535302392\
2864124682714495322759578165633421391609297722696178782065805102272979\
0471757521616178805294850768423898290193635176600276150984322616466736\
21262056104958269159031539952386366425328743319210680 qx qz + 
 810404710453684404449711145642531979844039181424682790052988605948928\
4686422846578071322066066720631974338178371081463783932655933964481466\
9811055583542474886853184796791286727893321546450269470111277027868266\
8922291528822664229437063866864574626411960357222442866546783767880330\
91167675047232878995040262731053202519064508242400 qy qz + 
 152830097494642265504510707250222331142777355310815085217970902751603\
0347034722765085422348829892611273758147588297336587513397169685986778\
0904949164516520452330880016462926479443433587855015411345385880175023\
6187275276468709349327131656991739575588792542630056334796338102103746\
0421261195208842266193695005452058189938179360184840 qz^2
List @@ (expr /. _?NumericQ -> 1)
MonomialList[expr] /. _Integer -> 1 // Sort

The result is:

{1, 2 c1, 2 c2, c1 c2, 2 c3, c1 c3, c2 c3, px, c3 px, py, c2 py, 
 c3 py, pz, c3 pz, qx, c2 qx, c3 qx, px qx, py qx, pz qx, 2 qy, c2 qy,
  c3 qy, px qy, py qy, pz qy, qx qy, 2 qz, c2 qz, c3 qz, px qz, py qz,
  pz qz, qx qz, qy qz}

But the true result I want is:
{1, c1, c1^2, c2, c1 c2, c2^2, c3, c1 c3, c2 c3, c3^2, px, c3 px, py,
c2 py, c3 py, pz, c3 pz, qx, c2 qx, c3 qx, px qx, py qx, pz qx, qy,
c2 qy, c3 qy, px qy, py qy, pz qy, qx qy, qy^2, qz, c2 qz, c3 qz,
px qz, py qz, pz qz, qx qz, qy qz, qz^2}

How to solve it?

5

You can actually state the problem without the lengthy input. Simply, your problem is like how to extract the symbolic terms of

expr = 3 a^2 + 2 a b + a

in which you have the square term, mixed symbol term and a single symbol term. You can use Apply function to extract each term in the summation

List@@expr

For each element, you can map this rule to them (y is ahead of three Blank[])

List @@ expr /. HoldPattern[Times[x_, y___]] -> {x, y}

Then you have

{{3, a^2}, {2, a, b}, c}

Then you can write a function to handle the lists:

f[x_List] := Times @@ Rest[x]
f[x_Numeric] := x
f[x_Symbol] := x

As you map it to the list, you get

In[]:= f /@{{3, a^2}, {2, a, b}, c}
Out[]= {a^2, a b, c}
6

this code may help you.

Level[expr // Expand, 1][[2 ;;, 2 ;;]]

out form is :

{c1,c1^2,c2,c1 c2,c2^2,c3,c1 c3,c2 c3,c3^2,px,c3 px,py,c2 py,c3 py,pz,c3 pz,qx,c2 qx,c3 qx,px qx,py qx,pz qx,qy,c2 qy,c3 qy,px qy,py qy,pz qy,qx qy,qy^2,qz,c2 qz,c3 qz,px qz,py qz,pz qz,qx qz,qy qz,qz^2}
7

Thanks to Mr.wayne wang and Mr. Shenghui Yang very much. With your help the problem can be solved.Once again, thanks to Mr. Simon Schmidt and Mr. Arnoud Buzing.