# inverse(RationalMap) -- inverse of a birational map

## Synopsis

• Function: inverse
• Usage:
inverse phi
inverse(phi,Certify=>b)
• Inputs:
• phi, , which has to be birational, and b is a boolean value, that is, true or false (the default value is false)
• Outputs:
• , the inverse map of phi

## Description

inverse(phi,Certify=>true) is the same as inverseMap(phi,Certify=>true,Verbose=>false), while inverse(phi,Certify=>false) applies a middle ground approach between inverseMap(phi,Certify=>true) and inverseMap(phi,Certify=>false). The procedure for the latter is as follows: It first computes the inverse map of phi using psi = inverseMap phi. Then it is checked that psi phi p == p and phi psi q == q, where p,q are, respectively, a random point on the source and the target of phi. Finally, if the tests pass, the command forceInverseMap(phi,psi) is invoked and psi is returned.

 i1 : R = QQ[x_0..x_4]; phi = rationalMap minors(4,random(R^{4:1},R^5)) -- special Cremona transformation of P^4 of type (4,4) o2 = -- rational map -- source: Proj(QQ[x , x , x , x , x ]) 0 1 2 3 4 target: Proj(QQ[x , x , x , x , x ]) 0 1 2 3 4 defining forms: { 87713 4 171940669 3 786693217 2 2 1176261 3 174575 4 456216499 3 52359194927 2 5579886061 2 13389751 3 191644459 2 2 59850847787 2 831631429 2 2 31307303 3 8534795951 3 10081913 4 11675519 3 192249615263 2 36291945199 2 56838053 3 1873054933 2 292082344411 23709080881 2 74483500913 2 35210041763 2 2080929049 3 203544727 2 2 5850860317 2 53160721 2 2 34691361079 2 10467532387 2 58824896633 2 2 398325457 3 514981 3 829687 3 109219 4 87928937 3 37380364421 2 12850450633 2 304700929 3 600713887 2 520240788881 12352174879 2 192049751 2 3302801963 2 36931661 3 588577417 2 1475796509 8966991391 2 47105896363 10761984361 1540753237 2 7830532057 2 10360739819 2 126855097379 2 97816669 3 194637089 2 2 129721589 2 810200723 2 2 3458531599 2 10016990171 2 402378101 2 2 210959377967 2 409370286581 2 374657652319 2 11558520289 2 2 862261831 3 190847047 3 211570741 3 4335215 3 489173 4 - -----x + ---------x x - ---------x x - -------x x + ------x - ---------x x + -----------x x x - ----------x x x - --------x x - ---------x x + -----------x x x - ---------x x + --------x x + ----------x x + --------x - --------x x + ------------x x x - -----------x x x + --------x x + ----------x x x + ------------x x x x - -----------x x x + -----------x x x + -----------x x x + ----------x x + ---------x x - ----------x x x - --------x x + -----------x x x - -----------x x x + -----------x x - ---------x x + ------x x + -------x x - ------x - --------x x + -----------x x x - -----------x x x - ---------x x - ---------x x x + ------------x x x x - -----------x x x + ---------x x x + ----------x x x + --------x x - ---------x x x + ----------x x x x - ----------x x x - -----------x x x x - -----------x x x x - ----------x x x + ----------x x x + -----------x x x + ------------x x x + --------x x - ---------x x + ---------x x x - ---------x x - ----------x x x + -----------x x x - ---------x x - ------------x x x + ------------x x x - ------------x x x + -----------x x - ---------x x + ---------x x - ---------x x + -------x x - ------x , 7840 0 3628800 0 1 16934400 0 1 8960 0 1 2268 1 21168000 0 2 444528000 0 1 2 39513600 0 1 2 282240 1 2 105840000 0 2 889056000 0 1 2 35562240 1 2 3307500 0 2 296352000 1 2 1764000 2 441000 0 3 1143072000 0 1 3 152409600 0 1 3 1306368 1 3 2222640000 0 2 3 1600300800 0 1 2 3 114307200 1 2 3 2222640000 0 2 3 5334336000 1 2 3 222264000 2 3 3175200 0 3 342921600 0 1 3 1224720 1 3 222264000 0 2 3 228614400 1 2 3 1333584000 2 3 4762800 0 3 30240 1 3 1058400 2 3 3780 3 19051200 0 4 304819200 0 1 4 152409600 0 1 4 7620480 1 4 127008000 0 2 4 3556224000 0 1 2 4 266716800 1 2 4 42336000 0 2 4 84672000 1 2 4 7056000 2 4 35721000 0 3 4 78382080 0 1 3 4 54867456 1 3 4 533433600 0 2 3 4 65318400 1 2 3 4 15876000 2 3 4 428652000 0 3 4 34292160 1 3 4 1333584000 2 3 4 6350400 3 4 11907000 0 4 12096000 0 1 4 15240960 1 4 63504000 0 2 4 1778112000 1 2 4 21168000 2 4 2857680000 0 3 4 2286144000 1 3 4 3333960000 2 3 4 190512000 3 4 19845000 0 4 63504000 1 4 3307500 2 4 254016 3 4 126000 4 77687 4 1893375163 3 745990081 2 2 246515089 3 35446399 4 12828001 3 4299402481 2 117657989077 2 520789147 3 47018753 2 2 2447714863 2 14479054193 2 2 536047 3 1192105381 3 518639 4 5114703379 3 3691001057 2 79302551 2 613515787 3 6101491049 2 2497012027 24426294703 2 8032480129 2 393103933 2 980054 3 1676997047 2 2 16148725991 2 541267781 2 2 35711433817 2 1583696827 2 1235700847 2 2 87106963 3 625319561 3 298242641 3 544463 4 13856723 3 10624110179 2 48549025723 2 734224931 3 409063541 2 1819317229 98792752963 2 274596281 2 94078309523 2 122592691 3 61593467 2 1430441815217 7429733591 2 83059865963 4387536922399 22245892841 2 161480778617 2 14301904817 2 22041159893 2 261691907 3 36414451 2 2 24541197491 2 1293610919 2 2 75958487 2 49534308121 2 789360553 2 2 138709401809 2 38760394277 2 99078815873 2 7078651 2 2 3013669 3 116122337 3 320231 3 17810837 3 1247 4 -----x - ----------x x - ---------x x + ---------x x + --------x - --------x x - ----------x x x - ------------x x x + ---------x x - --------x x - ----------x x x - -----------x x - ------x x - ----------x x - ------x + ----------x x + ----------x x x + --------x x x + ---------x x + ----------x x x + ----------x x x x + -----------x x x + ----------x x x + ---------x x x + ------x x - ----------x x - -----------x x x - ---------x x + -----------x x x - ----------x x x + ----------x x - --------x x - ---------x x + ---------x x + ------x + --------x x + -----------x x x + -----------x x x + ---------x x + ---------x x x + ----------x x x x + -----------x x x + ---------x x x + -----------x x x + ---------x x + ---------x x x - -------------x x x x - ----------x x x + -----------x x x x + -------------x x x x - -----------x x x - ------------x x x - -----------x x x - -----------x x x + ---------x x - --------x x + -----------x x x - ----------x x + --------x x x + -----------x x x + ---------x x - ------------x x x - -----------x x x - -----------x x x + -------x x + -------x x - ---------x x + ------x x + --------x x - ----x , 44100 0 74088000 0 1 3704400 0 1 1693440 0 1 80640 1 2940000 0 2 37044000 0 1 2 177811200 0 1 2 8467200 1 2 1470000 0 2 9878400 0 1 2 50803200 1 2 17640 0 2 14817600 1 2 117600 2 277830000 0 3 17781120 0 1 3 864000 0 1 3 907200 1 3 123480000 0 2 3 6945750 0 1 2 3 190512000 1 2 3 222264000 0 2 3 5556600 1 2 3 231525 2 3 20837250 0 3 190512000 0 1 3 10886400 1 3 666792000 0 2 3 142884000 1 2 3 66679200 2 3 4762800 0 3 8164800 1 3 9525600 2 3 113400 3 1984500 0 4 296352000 0 1 4 254016000 0 1 4 2419200 1 4 13608000 0 2 4 7056000 0 1 2 4 190512000 1 2 4 95256000 0 2 4 533433600 1 2 4 15876000 2 4 200037600 0 3 4 11430720000 0 1 3 4 11664000 1 3 4 1714608000 0 2 3 4 34292160000 1 2 3 4 444528000 2 3 4 1428840000 0 3 4 36288000 1 3 4 952560000 2 3 4 13608000 3 4 31752000 0 4 317520000 0 1 4 6048000 1 4 3402000 0 2 4 119070000 1 2 4 31752000 2 4 1428840000 0 3 4 81648000 1 3 4 476280000 2 3 4 38880 3 4 302400 0 4 756000 1 4 12096 2 4 168000 3 4 300 4 20470217 4 6197290213 3 142427891567 2 2 1607881319 3 402659 4 6075301 3 261895079099 2 1197196974853 2 129429707 3 4720865 2 2 16381014529 2 185942825261 2 2 522507949 3 50490451 3 3948499 4 187962641 3 9589727641 2 5095947877 2 8620609 3 242438299 2 30818449 2599354883 2 299855093 2 6354704233 2 30475577 3 35776819 2 2 155423077 2 331874303 2 2 9042325657 2 29069826377 2 75778393 2 2 4296577 3 33468533 3 15349 3 31001 4 7356550399 3 271122359 2 47203852511 2 404391767 3 87849878513 2 92563384943 476223629723 2 243843073 2 34868254657 2 16067797 3 253641949 2 102150887 1308957383 2 19959639787 225792765497 8331927001 2 2381334211 2 1895122339 2 42535616623 2 157432169 3 1457325563 2 2 48181219883 2 20798623 2 2 10545160561 2 119416726721 2 6035444641 2 2 916767947 2 2861857 2 1443756259 2 5256437 2 2 670106873 3 11389349 3 17933177 3 17828999 3 19613 4 --------x - ----------x x - ------------x x + ----------x x + ------x + -------x x - ------------x x x - -------------x x x + ---------x x - -------x x - -----------x x x - ------------x x - ---------x x - --------x x - -------x + ---------x x - ----------x x x + ----------x x x + -------x x + ---------x x x + --------x x x x + ----------x x x - ---------x x x + ----------x x x - --------x x + --------x x + ---------x x x - ---------x x - ----------x x x + -----------x x x - --------x x - -------x x + --------x x - -----x x - -----x + ----------x x + ---------x x x + -----------x x x + ---------x x + -----------x x x + -----------x x x x + ------------x x x - ---------x x x + -----------x x x + --------x x + ---------x x x + ---------x x x x - ----------x x x + -----------x x x x + ------------x x x x + ----------x x x + ----------x x x - ----------x x x - -----------x x x + ---------x x + ----------x x + -----------x x x - --------x x + -----------x x x + ------------x x x + ----------x x - ---------x x x - -------x x x - ----------x x x - -------x x + ---------x x + --------x x + --------x x - --------x x + -----x , 882000 0 42336000 0 1 169344000 0 1 4838400 0 1 720 1 705600 0 2 423360000 0 1 2 1016064000 0 1 2 403200 1 2 56448 0 2 26460000 0 1 2 508032000 1 2 7056000 0 2 288000 1 2 235200 2 2646000 0 3 63504000 0 1 3 14515200 0 1 3 20736 1 3 13230000 0 2 3 4704000 0 1 2 3 5443200 1 2 3 6615000 0 2 3 127008000 1 2 3 1512000 2 3 705600 0 3 345600 0 1 3 622080 1 3 127008000 0 2 3 130636800 1 2 3 1587600 2 3 151200 0 3 373248 1 3 19200 2 3 25920 3 158760000 0 4 15876000 0 1 4 63504000 0 1 4 2419200 1 4 1905120000 0 2 4 254016000 0 1 2 4 508032000 1 2 4 76204800 0 2 4 127008000 1 2 4 1814400 2 4 3969000 0 3 4 31752000 0 1 3 4 806400 1 3 4 381024000 0 2 3 4 1143072000 1 2 3 4 285768000 2 3 4 47628000 0 3 4 6531840 1 3 4 571536000 2 3 4 1632960 3 4 31752000 0 4 63504000 0 1 4 134400 1 4 63504000 0 2 4 127008000 1 2 4 63504000 2 4 3175200 0 3 4 6720 1 3 4 4762800 2 3 4 108864 3 4 5292000 0 4 302400 1 4 98000 2 4 226800 3 4 1800 4 1500391 4 8275752563 3 256341089 2 2 155007079 3 746581 4 97867211 3 164921515901 2 24403628009 2 888520553 3 382779157 2 2 5957276623 2 509959133 2 2 885643 3 25594339 3 60397 4 124434781 3 90257972921 2 1098062669 2 2057266549 3 2039769983 2 225727376089 2460325939 2 8187871753 2 10446122737 2 675949 3 151489834 2 2 887711869 2 165710771 2 2 432824251999 2 13643011097 2 2379244109 2 2 94290809 3 6889639 3 1258555709 3 336733 4 108579713 3 34623880909 2 1486648359659 2 3469044511 3 812299043 2 101676789773 7196517913 2 4944129079 2 111040487203 2 1960711 3 738628408877 2 29850049441 23161548511 2 350948383 500867067709 3587852269 2 35587222067 2 1638347143 2 4960572769 2 89034755 3 2840497253 2 2 185863846411 2 389117173 2 2 36324424369 2 17653930411 2 103864601 2 2 594188313743 2 2800420663 2 12401450617 2 898438523 2 2 1147560823 3 912182179 3 20621633 3 26614447 3 563 4 - -------x - ----------x x + ---------x x + ---------x x - ------x - --------x x - ------------x x x + -----------x x x + ---------x x - ---------x x - ----------x x x + ---------x x - ------x x - --------x x - -----x + ---------x x + -----------x x x + ----------x x x - ----------x x + ----------x x x + ------------x x x x + ----------x x x + ----------x x x + -----------x x x - -------x x + ---------x x + ---------x x x - ---------x x - ------------x x x + -----------x x x - ----------x x - --------x x - -------x x - ----------x x - ------x - ---------x x + -----------x x x + -------------x x x - ----------x x - ---------x x x - ------------x x x x + ----------x x x - ----------x x x - ------------x x x - -------x x + ------------x x x + -----------x x x x - -----------x x x + ---------x x x x + ------------x x x x + ----------x x x - -----------x x x + ----------x x x - ----------x x x + --------x x + ----------x x + ------------x x x + ---------x x - -----------x x x + -----------x x x - ---------x x - ------------x x x + ----------x x x - -----------x x x - ---------x x - ----------x x + ---------x x - --------x x - --------x x - ----x , 211680 0 88905600 0 1 3704400 0 1 5080320 0 1 6480 1 3528000 0 2 889056000 0 1 2 106686720 0 1 2 38102400 1 2 10584000 0 2 74088000 0 1 2 5292000 1 2 49000 0 2 1411200 1 2 12000 2 2778300 0 3 1333584000 0 1 3 31752000 0 1 3 5443200 1 3 22226400 0 2 3 833490000 0 1 2 3 63504000 1 2 3 222264000 0 2 3 90720000 1 2 3 3528000 2 3 10418625 0 3 63504000 0 1 3 604800 1 3 6667920000 0 2 3 114307200 1 2 3 119070000 2 3 4762800 0 3 466560 1 3 95256000 2 3 226800 3 95256000 0 4 889056000 0 1 4 5334336000 0 1 4 38102400 1 4 35721000 0 2 4 5715360000 0 1 2 4 381024000 1 2 4 95256000 0 2 4 952560000 1 2 4 84000 2 4 10001880000 0 3 4 228614400 0 1 3 4 228614400 1 3 4 3499200 0 2 3 4 1714608000 1 2 3 4 105840000 2 3 4 8001504000 0 3 4 76204800 1 3 4 381024000 2 3 4 3048192 3 4 272160000 0 4 889056000 0 1 4 3048192 1 4 1905120000 0 2 4 95256000 1 2 4 3024000 2 4 8001504000 0 3 4 28576800 1 3 4 190512000 2 3 4 38102400 3 4 63504000 0 4 12700800 1 4 604800 2 4 846720 3 4 2800 4 25197 4 140616709 3 13520392891 2 2 13683556193 3 43193719 4 1310991 3 13938392243 2 20490507127 2 1092604379 3 25878779 2 2 61718231 2 17113948007 2 2 3781009 3 1399587607 3 3213307 4 201997 3 4300057681 2 18208353301 2 2892598211 3 958803161 2 8509111559 95425957337 2 8879315339 2 123111557327 2 804820927 3 838659919 2 2 64213060183 2 30588325141 2 2 9844329091 2 3831602699 2 421910291 2 2 1356293 3 90385021 3 1080170969 3 240439 4 708417343 3 69211419139 2 821880229741 2 8234269157 3 17527342831 2 5079418246169 271977807167 2 3766010893 2 73836492767 2 6083219 3 69268081709 2 166294514647 188202892601 2 30929665997 15654734391433 584160397 2 590797374319 2 395965971767 2 491764815781 2 1214588279 3 2308919491 2 2 437921877679 2 40210722691 2 2 2916578449 2 395005523113 2 315679271 2 2 534410724893 2 69884537 2 216151621817 2 2988196411 2 2 97251001 3 848159597 3 194830663 3 276462167 3 52207 4 - -----x - ---------x x + -----------x x + -----------x x - --------x - -------x x + -----------x x x + -----------x x x + ----------x x + --------x x + --------x x x - -----------x x - -------x x - ----------x x - -------x - ------x x + ----------x x x + -----------x x x - ----------x x + ---------x x x + ----------x x x x + -----------x x x + ----------x x x + ------------x x x + ---------x x - ---------x x - -----------x x x - -----------x x - ----------x x x + ----------x x x - ---------x x + -------x x - --------x x + ----------x x + ------x - ---------x x - -----------x x x + ------------x x x + ----------x x - -----------x x x - -------------x x x x + ------------x x x - ----------x x x - -----------x x x - -------x x + -----------x x x + ------------x x x x + ------------x x x + -----------x x x x + --------------x x x x + ---------x x x - ------------x x x - ------------x x x - ------------x x x + ----------x x - ----------x x - ------------x x x + -----------x x - ----------x x x - ------------x x x - ---------x x + ------------x x x - --------x x x + ------------x x x + ----------x x - --------x x - ---------x x - ---------x x + ---------x x - -----x 2800 0 6350400 0 1 50803200 0 1 33868800 0 1 181440 1 196000 0 2 381024000 0 1 2 47628000 0 1 2 10160640 1 2 2268000 0 2 2126250 0 1 2 762048000 1 2 992250 0 2 31752000 1 2 529200 2 73500 0 3 28576800 0 1 3 32659200 0 1 3 6531840 1 3 15876000 0 2 3 21432600 0 1 2 3 137168640 1 2 3 158760000 0 2 3 571536000 1 2 3 95256000 2 3 15876000 0 3 228614400 0 1 3 65318400 1 3 190512000 0 2 3 95256000 1 2 3 31752000 2 3 352800 0 3 604800 1 3 31752000 2 3 15120 3 47628000 0 4 444528000 0 1 4 4267468800 0 1 4 152409600 1 4 714420000 0 2 4 16003008000 0 1 2 4 1524096000 1 2 4 95256000 0 2 4 533433600 1 2 4 211680 2 4 1000188000 0 3 4 2667168000 0 1 3 4 457228800 1 3 4 240045120 0 2 3 4 48009024000 1 2 3 4 14817600 2 3 4 4000752000 0 3 4 1524096000 1 3 4 2667168000 2 3 4 27216000 3 4 190512000 0 4 1778112000 0 1 4 304819200 1 4 47628000 0 2 4 1333584000 1 2 4 5292000 2 4 4000752000 0 3 4 71442000 1 3 4 1333584000 2 3 4 95256000 3 4 3969000 0 4 127008000 1 4 5292000 2 4 21168000 3 4 28000 4 } o2 : RationalMap (rational map from PP^4 to PP^4) i3 : time inverse phi -- used 0.150425 seconds o3 = -- rational map -- source: Proj(QQ[x , x , x , x , x ]) 0 1 2 3 4 target: Proj(QQ[x , x , x , x , x ]) 0 1 2 3 4 defining forms: { 4 3 2 2 3 4 3 2 2 3 2 2 2 2 2 3 3 4 3 2 2 3 2 2 2 2 3 2 2 2 2 2 2 2 2 2 3 3 3 4 3 2 2 3 2 2 2 2 3 2 2 2 2 2 2 3 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 4 - 122896266822000x + 610226014948200x x - 576753063780600x x + 102720879015300x x - 9414842673600x + 3699852055200x x + 294739615170000x x x - 138153302961660x x x - 20531975385600x x + 67442944887360x x + 72946285062840x x x - 22536566332560x x - 20227694518080x x + 53474573648160x x - 19027177344000x + 543647816472600x x - 729632501047800x x x + 393880976764800x x x + 17332571210850x x + 138326986841400x x x - 239573496489940x x x x - 50612943405150x x x + 76933021250736x x x - 76938251041620x x x + 61524106237440x x + 388020756853800x x - 454982331858600x x x + 275439328969350x x + 335496966653240x x x - 334524418004700x x x + 74995973730720x x - 571931229844800x x + 411337054789500x x - 177295703054400x x - 72053361828000x - 470991675558600x x + 684524587950000x x x - 343387979006100x x x + 111129455341800x x - 52885876241520x x x + 227503935136240x x x x - 265242788438040x x x + 21164925227040x x x + 182800484601840x x x - 42083923896000x x + 998536378213080x x x - 1064263857132840x x x x + 603623670792450x x x + 676716868932748x x x x - 626804085414660x x x x + 163725370837560x x x - 160427754360720x x x + 340388071535400x x x - 22401861485640x x x - 3623047936200x x - 426737044818000x x + 227827854626400x x x - 49334139653400x x - 92973815620560x x x + 10918769215200x x x + 26397954424800x x + 757906867892160x x x - 153494997056100x x x + 17739208087440x x x + 50625488377800x x - 271153558821600x x + 8845812435600x x + 7986790425600x x + 42802362262800x x - 42149700432000x , 0 0 1 0 1 0 1 1 0 2 0 1 2 0 1 2 1 2 0 2 0 1 2 1 2 0 2 1 2 2 0 3 0 1 3 0 1 3 1 3 0 2 3 0 1 2 3 1 2 3 0 2 3 1 2 3 2 3 0 3 0 1 3 1 3 0 2 3 1 2 3 2 3 0 3 1 3 2 3 3 0 4 0 1 4 0 1 4 1 4 0 2 4 0 1 2 4 1 2 4 0 2 4 1 2 4 2 4 0 3 4 0 1 3 4 1 3 4 0 2 3 4 1 2 3 4 2 3 4 0 3 4 1 3 4 2 3 4 3 4 0 4 0 1 4 1 4 0 2 4 1 2 4 2 4 0 3 4 1 3 4 2 3 4 3 4 0 4 1 4 2 4 3 4 4 4 3 2 2 3 4 3 2 2 3 2 2 2 2 2 3 3 4 3 2 2 3 2 2 2 2 3 2 2 2 2 2 2 2 2 2 3 3 3 4 3 2 2 3 2 2 2 2 3 2 2 2 2 2 2 3 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 4 15912286185600x - 32666729760000x x - 15108013053000x x + 15508549507500x x - 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