Singular Manual: locNormal

D.4.24.2 locNormal

Procedure from library normal.lib (see normal_lib).

Usage:: locNormal(I [,options]); I = prime ideal, options = list of options.
Optional parameters in list options (can be entered in any order):
modular: use a modular approach for the local computations. The number of primes is increased one at a time, starting with 2 primes, until the result stabilizes.
noVerification: if the modular approach is used, the result will not be verified.
Assume:: I is a prime ideal (the algorithm will also work for radical ideals as long as the normal command does not detect that the ideal under consideration is not prime).
Return:: a list of an ideal U and a universal denominator d such that U/d is the normalization.
Remarks:: We use the local-to-global algorithm given in [1] to compute the normalization of A = R/I, where R is the basering.
The idea is to stratify the singular locus of A, apply the normalization algorithm given in [2] locally at each stratum, and put the local results together.
If the option modular is given, the result is returned as a probabilistic result or verified, depending on whether the option noVerification is used or not.
The normalization of A is represented as an R-module by returning a list of U and d, where U is an ideal of A and d is an element of A such that U/d is the normalization of A. In fact, U and d are returned as an ideal and a polynomial of the base ring R.
References:: [1] Janko Boehm, Wolfram Decker, Santiago Laplagne, Gerhard Pfister, Stefan Steidel, Andreas Steenpass: Parallel algorithms for normalization, http://arxiv.org/abs/1110.4299, 2011.
[2] Gert-Martin Greuel, Santiago Laplagne, Frank Seelisch: Normalization of Rings, Journal of Symbolic Computation 9 (2010), p. 887-901

Example:

LIB "normal.lib";
ring R = 0,(x,y,z),dp;
int k = 4;
poly f = (x^(k+1)+y^(k+1)+z^(k+1))^2-4*(x^(k+1)*y^(k+1)+y^(k+1)*z^(k+1)+z^(k+1)*x^(k+1));
f = subst(f,z,3x-2y+1);
ring S = 0,(x,y),dp;
poly f = imap(R,f);
ideal i = f;
list L = locNormal(i);
L;
==> [1]:
==>    _[1]=x5y4-405/122x4y5+270/61x3y6-180/61x2y7+60/61xy8-33/244y9+405/244x\
   4y4-270/61x3y5+270/61x2y6-120/61xy7+20/61y8+135/122x3y4-135/61x2y5+90/61x\
   y6-20/61y7+45/122x2y4-30/61xy5+10/61y6+15/244xy4-5/122y5+1/244y4
==>    _[2]=x4y6-1280612/517599x3y7+129960/57511x2y8-51921/57511xy9+15939/115\
   022y10-x4y5+640306/172533x3y6-259920/57511x2y7+130260/57511xy8-93219/2300\
   44y9+1/4x4y4-320153/172533x3y5+194940/57511x2y6-130260/57511xy7+28980/575\
   11y8+320153/1035198x3y4-64980/57511x2y5+65130/57511xy6-19320/57511y7+1624\
   5/115022x2y4-32565/115022xy5+7245/57511y6+6513/230044xy4-1449/57511y5+483\
   /230044y4
==>    _[3]=x7y3-198131947/41752986x3y7+276768011/41752986x2y8-196444003/6262\
   9479xy9+2940134/5693589y10-1/2x7y2+1457/363x6y3-120075/7381x4y5+143365570\
   20/424488691x3y6-54708132437/1697954764x2y7+54940261271/3820398219xy8-553\
   63533761/22922389314y9-1457/1452x6y2+11177/2178x5y3+48705/14762x4y4-11666\
   788930/424488691x3y5+15900606780/424488691x2y6-105451349237/5093864292xy7\
   +92997673871/22922389314y8-11177/13068x5y2+390/121x4y3+5188408600/1273466\
   073x3y4-6788011830/424488691x2y5+16943306620/1273466073xy6-156194566037/4\
   5844778628y7-195/484x4y2+25/22x3y3+788905705/424488691x2y4-1659910480/424\
   488691xy5+5821552180/3820398219y6-5/44x3y2+28/121x2y3+948269669/254693214\
   6xy4-437697131/1273466073y5-7/363x2y2+28/1089xy3+632707387/22922389314y4-\
   2/1089xy2+4/3267y3-1/13068y2
==>    _[4]=x8y2-11171668429/2526055653x3y7+10631780685/1684037102x2y8-252160\
   2135/842018551xy9+36974970/76547141y10+131355/14641x6y3-40695195/893101x4\
   y5+4254040854900/51363131611x3y6-13830480846585/205452526444x2y7+13524033\
   23595/51363131611xy8-415006693815/102726263222y9-131355/58564x6y2+437625/\
   29282x5y3+13118925/1786202x4y4-3619207734770/51363131611x3y5+454036291110\
   0/51363131611x2y6-9155055414195/205452526444xy7+829542368865/102726263222\
   y8-145875/58564x5y2+150690/14641x4y3+1396264546400/154089394833x3y4-20032\
   84117350/51363131611x2y5+1577081427300/51363131611xy6-1522216727355/20545\
   2526444y7-75345/58564x4y2+10015/2662x3y3+212182593525/51363131611x2y4-475\
   908659100/51363131611xy5+178766109900/51363131611y6-2003/5324x3y2+11340/1\
   4641x2y3+84987619515/102726263222xy4-41195775465/51363131611y5-945/14641x\
   2y2+1260/14641xy3+6299623005/102726263222y4-90/14641xy2+60/14641y3-15/585\
   64y2
==>    _[5]=xy10-13641993431/29843836298y11+46739291489040/14068433355199x3y7\
   -1402422957480/2707174872767x2y8-2726350272361741/858174434667139xy9+1611\
   8871666481117/10298093216005668y10-19651963320/244621609x7y2+481995698818\
   8/14921918149x6y3-4772003774302899/3640948028356x4y5+11231741615995970592\
   3/52348640514695479x3y6-79937423352024474960/52348640514695479x2y7+276063\
   32976408874538/52348640514695479xy8-47304951053682762193/6281836861763457\
   48y9-2407189841458/14921918149x6y2+24668834517872/44765754447x5y3+4928695\
   11159993/3640948028356x4y4-194013862593198827445/104697281029390958x3y5+1\
   14730080433139187720/52348640514695479x2y6-53952309350804665753/523486405\
   14695479xy7+27829401145126589552/157045921544086437y8-2053170342108/14921\
   918149x5y2+5815150716825/14921918149x4y3+17773939452046933761/10469728102\
   9390958x3y4-101673832865669738595/104697281029390958x2y5+7784130709571461\
   4895/104697281029390958xy6-18164179800006473511/104697281029390958y7-1935\
   790816725/29843836298x4y2+2188915306380/14921918149x3y3+82439267253529385\
   25/104697281029390958x2y4-46885347439431687417/209394562058781916xy5+4385\
   750100429190396/52348640514695479y6-273225499965/14921918149x3y2+46475535\
   0780/14921918149x2y3+3370391526053507673/209394562058781916xy4-1205182872\
   7946988179/628183686176345748y5-46406394930/14921918149x2y2+53160990664/1\
   4921918149xy3+768558233460398801/628183686176345748y4-8846721193/29843836\
   298xy2+7757445871/44765754447y3-368853453/29843836298y2
==>    _[6]=x2y9-3114342649/14921918149y11+71715552636640/14068433355199x3y7-\
   10326355425840/2707174872767x2y8-27438576677503/2574523304001417xy9+12343\
   62165817957/1716348869334278y10-32130825936/244621609x7y2+6806922686376/1\
   4921918149x6y3-3029593984301673/1820474014178x4y5+139043949507800124594/5\
   2348640514695479x3y6-97058322862423257696/52348640514695479x2y7+329115389\
   32090793916/52348640514695479xy8-82773301884415654015/942275529264518622y\
   9-11270379846532/44765754447x6y2+32430188239264/44765754447x5y3+454618548\
   330651/1820474014178x4y4-125216469593410993359/52348640514695479x3y5+1422\
   84514069585602960/52348640514695479x2y6-196066072422479493922/15704592154\
   4086437xy7+99289741986726605552/471137764632259311y8-27248580046120/13429\
   7263341x5y2+7012104958710/14921918149x4y3+15990354996340180195/5234864051\
   4695479x3y4-66097449392376043665/52348640514695479x2y5+481757523558892804\
   45/52348640514695479xy6-98556701006296801525/471137764632259311y7-1332939\
   709935/14921918149x4y2+2380960733640/14921918149x3y3+7224959158180586787/\
   52348640514695479x2y4-30467013227952736155/104697281029390958xy5+16197489\
   000582285164/157045921544086437y6-346896756750/14921918149x3y2+4493049386\
   00/14921918149x2y3+8590747340076337697/314091843088172874xy4-777467505650\
   5118917/314091843088172874y5-53690789868/14921918149x2y2+136526815504/447\
   65754447xy3+1885693976900761535/942275529264518622y4-4644106987/149219181\
   49xy2+18085098926/134297263341y3-1604660495/134297263341y2
==>    _[7]=x3y8-1882380060/14921918149y11+105584615807039/28136866710398x3y7\
   -5309394350400/2707174872767x2y8+106130908873980/858174434667139xy9+37786\
   2820904250/858174434667139y10-75342805857/489243218x7y2+14521778847783/29\
   843836298x6y3-2967743903660235/1820474014178x4y5+133202341074203253618/52\
   348640514695479x3y6-365914149460772023449/209394562058781916x2y7+61220670\
   222754103643/104697281029390958xy8-16872519516391121949/20939456205878191\
   6y9-17138559076263/59687672596x6y2+10907993961207/14921918149x5y3+1135344\
   139166295/3640948028356x4y4-124450152591508141586/52348640514695479x3y5+1\
   36568918715665449950/52348640514695479x2y6-245792134725795600003/20939456\
   2058781916xy7+10216432836746627247/52348640514695479y8-13315720121847/596\
   87672596x5y2+13096312959645/29843836298x4y3+39459847798685297699/10469728\
   1029390958x3y4-66113796664298888955/52348640514695479x2y5+461737807660470\
   44790/52348640514695479xy6-41009835106162224747/209394562058781916y7-5581\
   916065215/59687672596x4y2+1988439910470/14921918149x3y3+17598598904655245\
   925/104697281029390958x2y4-15236039047367783700/52348640514695479xy5+5150\
   621717328561750/52348640514695479y6-1359163478835/59687672596x3y2+3185739\
   33930/14921918149x2y3+6863608640920827375/209394562058781916xy4-257683948\
   2472381245/104697281029390958y5-48077105553/14921918149x2y2+25595308152/1\
   4921918149xy3+491807979553076973/209394562058781916y4-3709858806/14921918\
   149xy2+1737520137/29843836298y3-505672911/59687672596y2
==>    _[8]=y12-792866177120/567766754489y11+71739107903344684360/75787227290\
   224576159x3y7-6776196114325803340/239076426783042827x2y8+5660051028866160\
   747560/227361681870673728477xy9-1378378888306420096120/227361681870673728\
   477y10+54080943679608987920/75832631029814307x7y2-20372331336611010626720\
   /9175748354607531147x6y3+124539523923485536574400/16961231807001799999x4y\
   5-122845899372102295480068591360/10730031426977285717167379x3y6+254160278\
   504808008196422977310/32190094280931857151502137x2y7-23457553239663972158\
   298661520/8779116622072324677682401xy8+108434060005029624246726291700/289\
   710848528386714363519233y9+36670159154917215208210/27527245063822593441x6\
   y2-90770857096936328196140/27527245063822593441x5y3-287063666645095539961\
   050/186573549877019799989x4y4+10613954135676945444768543720/9754574024524\
   80519742489x3y5-127276313949227788472132569200/10730031426977285717167379\
   x2y6+515889037027564875254252318980/96570282842795571454506411xy7-2354694\
   5012696667764891375735/26337349866216974033047203y8+853900277997499310307\
   86/82581735191467780323x5y2-5896475011981357460190/3058582784869177049x4y\
   3-20270066639938379222191292340/10730031426977285717167379x3y4+5727114621\
   123780768952043040/975457402452480519742489x2y5-4349936643447689774393400\
   7880/10730031426977285717167379xy6+520824011585340698989966989385/5794216\
   97056773428727038466y7+2633363963748453360765/6117165569738354098x4y2-168\
   5315809747054639670/3058582784869177049x3y3-27662813514900838753146825680\
   /32190094280931857151502137x2y4+33251325513630703008741680/24184894275681\
   335200227xy5-44174045540608607665526736280/96570282842795571454506411y6+3\
   14495543808556642805/3058582784869177049x3y2-1100804445413356160/14495653\
   008858659x2y3-16590735631202580399939012640/96570282842795571454506411xy4\
   +1038553422090950622352716560/8779116622072324677682401y5+424969910646917\
   10980/3058582784869177049x2y2-11731499580868573125/3058582784869177049xy3\
   -3682717414300822201563051710/289710848528386714363519233y4+2518715232280\
   21415/260921754159455862xy2+10920872929239710/391382631239183793y3+204813\
   44169674981/782765262478367586y2
==> [2]:
==>    x5y4-405/122x4y5+270/61x3y6-180/61x2y7+60/61xy8-33/244y9+405/244x4y4-2\
   70/61x3y5+270/61x2y6-120/61xy7+20/61y8+135/122x3y4-135/61x2y5+90/61xy6-20\
   /61y7+45/122x2y4-30/61xy5+10/61y6+15/244xy4-5/122y5+1/244y4