degreeMap -- degree of a rational map between projective varieties

Synopsis

• Usage:
degreeMap phi
• Inputs:
• phi, , which represents a rational map $\Phi$ between projective varieties
• Optional inputs:
• BlowUpStrategy => ..., default value Eliminate,
• MathMode => ..., default value false, whether to ensure correctness of output
• Verbose => ..., default value true,
• Outputs:
• an integer, the degree of $\Phi$. So this value is 1 if and only if the map is birational onto its image.

Description

One important case is when $\Phi:\mathbb{P}^n=Proj(K[x_0,\ldots,x_n]) \dashrightarrow \mathbb{P}^m=Proj(K[y_0,\ldots,y_m])$ is a rational map between projective spaces, corresponding to a ring map $\phi$. If $p$ is a general point of $\mathbb{P}^n$, denote by $F_p(\Phi)$ the closure of $\Phi^{-1}(\Phi(p))\subseteq \mathbb{P}^n$. The degree of $\Phi$ is defined as the degree of $F_p(\Phi)$ if $dim F_p(\Phi) = 0$ and $0$ otherwise. If $\Phi$ is defined by forms $F_0(x_0,\ldots,x_n),\ldots,F_m(x_0,\ldots,x_n)$ and $I_p$ is the ideal of the point $p$, then the ideal of $F_p(\Phi)$ is nothing but the saturation ${(\phi(\phi^{-1}(I_p))):(F_0,....,F_m)}^{\infty}$.

 i1 : -- Take a rational map phi:P^8--->G(1,5) subset P^14 defined by the maximal minors -- of a generic 2 x 6 matrix of linear forms on P^8 (thus phi is birational onto its image) K=ZZ/3331; ringP8=K[x_0..x_8]; ringP14=K[t_0..t_14]; i4 : phi=map(ringP8,ringP14,gens minors(2,matrix pack(6,for i to 11 list random(1,ringP8)))) 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 o4 = map (ringP8, ringP14, {- 95x + 181x x + 1028x - 1384x x - 1455x x + 559x - 502x x + 1264x x - 162x x + 1209x - 180x x - 504x x - 1168x x - 676x x + 501x + 73x x + 1263x x + 1035x x + 844x x + 1593x x + 785x + 982x x - 412x x + 1335x x + 1136x x + 826x x + 1078x x + 1158x + 335x x - 982x x - 1479x x - 15x x + 1363x x + 1397x x - 575x x - 71x + 1255x x - 1138x x - 1590x x + 604x x + 1182x x - 63x x - 1382x x - 1255x x - 613x , - 1444x + 575x x + 767x - 1495x x + 1631x x - 217x - 294x x - 1511x x - 504x x - 1284x - 1459x x + 152x x + 141x x - 10x x - 95x + 1056x x + 654x x + 1397x x - 930x x + 578x x - 696x + 759x x + 733x x + 505x x - 609x x + 526x x - 659x x + 846x + 1253x x - 1519x x + 635x x + 576x x + 54x x - 1261x x - 822x x - 257x - 986x x + 356x x - 1488x x - 1561x x - 850x x - 85x x - 1350x x - 783x x - 1335x , - 871x + 1006x x - 1399x - 1636x x - 699x x - 769x - 307x x - 1645x x - 502x x - 719x + 1405x x + 870x x - 1133x x + 425x x - 1203x - 1601x x + 117x x - 382x x + 318x x - 117x x - 560x + 1135x x + 1468x x + 869x x - 943x x - 335x x - 1218x x + 201x - 11x x + 540x x - 710x x - 489x x + 1605x x + 1663x x - 423x x + 1246x + 97x x - 644x x + 1655x x + 1219x x + 1476x x + 1355x x + 1594x x + 893x x + 1150x , - 143x + 1240x x - 1042x + 1649x x + 1024x x + 794x + 1442x x - 1263x x + 537x x - 82x - 734x x - 1569x x - 798x x - 366x x + 1289x - 569x x - 254x x + 237x x - 1234x x - 807x x + 264x - 202x x - 616x x + 44x x + 1465x x + 685x x + 1630x x - 406x - 123x x - 4x x + 1583x x + 1235x x + 162x x + 1034x x - 1035x x + 737x + 660x x + 1459x x - 359x x - 1291x x + 1638x x - 325x x - 631x x + 73x x - 1471x , - 1340x + 31x x - 994x - 880x x - 89x x + 574x + 760x x - 1054x x + 772x x - 239x - 443x x + 1240x x + 637x x - 1423x x + 320x - 1363x x - 1139x x - 158x x - 325x x - 1578x x + 32x + 695x x + 305x x + 1012x x + 1492x x + 1290x x + 1579x x - 342x - 83x x - 104x x + 998x x - 92x x + 1554x x + 201x x - 237x x + 160x - 228x x - 543x x - 1147x x - 376x x + 1313x x + 603x x + 106x x - 1361x x + 699x , - 228x - 1510x x + 277x - 4x x - 22x x - 1526x + 234x x + 969x x + 1253x x - 1426x - 1474x x + 947x x + 194x x - 316x x - 988x - 1211x x + 1087x x + 536x x - 491x x + 870x x - 659x + 1490x x - 469x x + 1190x x + 807x x + 650x x + 448x x - 1353x - 218x x + 759x x - 253x x + 830x x - 1080x x - 143x x - 1313x x - 374x - 180x x + 741x x + 742x x - 1254x x + 458x x - 345x x + 597x x + 1567x x - 31x , 1120x + 709x x - 1538x - 1048x x - 162x x - 1518x - 73x x + 380x x + 533x x - 286x + 1374x x - 74x x - 22x x + 1535x x - 1071x - 839x x - 560x x + 928x x + 335x x - 1008x x + 810x - 448x x - 357x x - 107x x + 40x x + 784x x - 1423x x + 1276x + 147x x + 443x x - 598x x - 1077x x - 1214x x + 322x x - 1408x x + 72x - 63x x - 1513x x - 791x x + 11x x + 77x x + 836x x - 1100x x + 1637x x - 788x , 1331x + 318x x - 704x + 51x x + 275x x + 1149x + 1526x x + 768x x + 414x x - 782x - 262x x + 686x x - 380x x + 1377x x + 1077x + 1650x x - 1129x x - 508x x + 846x x + 1513x x + 460x - 1626x x - 1024x x + 862x x + 1352x x - 188x x - 1382x x - 650x + 55x x - 326x x + 1037x x + 705x x - 667x x + 1483x x + 1661x x - 1652x - 1052x x - 692x x - 542x x + 162x x + 582x x - 1369x x + 934x x + 1392x x + 1227x , - 346x + 1408x x - 1225x - 1536x x - 1028x x - 985x - 210x x - 1312x x + 915x x + 1633x - 202x x - 1636x x - 1653x x - 480x x - 1260x - 813x x - 1623x x - 1429x x + 1094x x - 747x x + 955x + 898x x - 795x x - 35x x - 566x x + 1631x x - 324x x + 926x - 132x x - 9x x - 1290x x - 543x x + 902x x + 735x x - 342x x - 400x + 900x x - 463x x + 694x x - 1262x x - 1449x x - 448x x - 1402x x - 731x x - 996x , 301x + 166x x - 955x - 739x x - 1199x x - 319x + 1047x x - 532x x + 902x x + 1195x - 663x x + 1215x x - 534x x - 332x x - 973x + 772x x - 308x x + 315x x - 454x x - 483x x - 239x - 1313x x - 419x x - 1340x x - 1388x x - 1340x x - 1665x x - 333x - 465x x - 1084x x + 676x x - 1612x x - 288x x + 11x x - 1170x x - 189x + 498x x - 889x x + 693x x + 1460x x - 473x x - 414x x - 122x x - 1659x x - 1421x , 14x - 1049x x + 1506x + 1235x x + 642x x - 1034x + 460x x + 150x x + 760x x - 1246x - 1407x x + 1570x x + 1403x x - 1610x x - 431x + 574x x + 893x x - 657x x + 417x x + 1362x x + 224x + 268x x + 1097x x + 1132x x + 148x x + 1331x x - 77x x - 756x + 228x x + 136x x - 1484x x - 1478x x - 13x x + 1620x x - 701x x - 769x - 760x x - 492x x - 1077x x - 1249x x - 834x x - 395x x - 1358x x - 988x x + 113x , - 1634x - 13x x + 805x - 21x x - 1655x x + 1479x - 1510x x - 646x x + 225x x - 1411x + 1227x x - 1108x x + 1291x x - 59x x - 142x + 586x x - 676x x + 655x x - 1476x x + 453x x - 1076x - 1152x x + 1373x x - 1191x x - 416x x + 699x x + 317x x + 825x - 1560x x - 488x x - 1035x x - 1561x x - 644x x - 1178x x - 1320x x + 158x + 889x x + 1444x x - 1486x x - 1211x x + 1269x x - 1228x x + 568x x + 1591x x + 1207x , 105x - 538x x - 1222x - 277x x + 716x x - 1067x - 428x x + 154x x - 469x x + 77x + 538x x - 179x x + 921x x - 223x x + 1093x - 262x x + 1299x x + 631x x + 1486x x - 1280x x - 121x - 50x x - 978x x - 694x x - 531x x + 505x x + 1412x x - 1061x + 1202x x + 448x x - 187x x + 1276x x - 121x x + 1361x x + 697x x + 682x + 1592x x + 705x x - 227x x - 7x x - 1423x x - 1446x x - 1578x x + 1511x x + 917x , 1270x - 391x x - 1116x - 287x x + 653x x + 1643x + 1623x x + 514x x - 14x x - 90x + 1232x x - 1434x x + 1296x x + 1522x x + 136x - 623x x - 607x x + 18x x + 896x x - 29x x + 1059x - 1053x x + 1643x x + 1652x x - 1190x x - 1073x x + 1470x x - 944x - 93x x - 187x x - 994x x - 1415x x - 229x x - 796x x + 1642x x + 1600x - 344x x + 905x x + 1032x x - 538x x - 891x x + 1243x x + 1290x x + 490x x - 1148x , 1613x + 175x x - 1346x - 1000x x - 1217x x - 729x - 1296x x + 1456x x + 745x x + 539x + 525x x - 811x x + 753x x + 1362x x + 1629x - 840x x + 513x x + 429x x + 842x x + 1414x x - 308x + 1415x x - 1461x x - 1135x x + 701x x + 766x x + 785x x + 1503x + 147x x + 929x x - 1220x x - 853x x + 493x x + 226x x + 1416x x + 280x - 7x x + 1632x x + 520x x + 1259x x + 157x x + 1596x x + 655x x - 42x x - 586x }) 0 0 1 1 0 2 1 2 2 0 3 1 3 2 3 3 0 4 1 4 2 4 3 4 4 0 5 1 5 2 5 3 5 4 5 5 0 6 1 6 2 6 3 6 4 6 5 6 6 0 7 1 7 2 7 3 7 4 7 5 7 6 7 7 0 8 1 8 2 8 3 8 4 8 5 8 6 8 7 8 8 0 0 1 1 0 2 1 2 2 0 3 1 3 2 3 3 0 4 1 4 2 4 3 4 4 0 5 1 5 2 5 3 5 4 5 5 0 6 1 6 2 6 3 6 4 6 5 6 6 0 7 1 7 2 7 3 7 4 7 5 7 6 7 7 0 8 1 8 2 8 3 8 4 8 5 8 6 8 7 8 8 0 0 1 1 0 2 1 2 2 0 3 1 3 2 3 3 0 4 1 4 2 4 3 4 4 0 5 1 5 2 5 3 5 4 5 5 0 6 1 6 2 6 3 6 4 6 5 6 6 0 7 1 7 2 7 3 7 4 7 5 7 6 7 7 0 8 1 8 2 8 3 8 4 8 5 8 6 8 7 8 8 0 0 1 1 0 2 1 2 2 0 3 1 3 2 3 3 0 4 1 4 2 4 3 4 4 0 5 1 5 2 5 3 5 4 5 5 0 6 1 6 2 6 3 6 4 6 5 6 6 0 7 1 7 2 7 3 7 4 7 5 7 6 7 7 0 8 1 8 2 8 3 8 4 8 5 8 6 8 7 8 8 0 0 1 1 0 2 1 2 2 0 3 1 3 2 3 3 0 4 1 4 2 4 3 4 4 0 5 1 5 2 5 3 5 4 5 5 0 6 1 6 2 6 3 6 4 6 5 6 6 0 7 1 7 2 7 3 7 4 7 5 7 6 7 7 0 8 1 8 2 8 3 8 4 8 5 8 6 8 7 8 8 0 0 1 1 0 2 1 2 2 0 3 1 3 2 3 3 0 4 1 4 2 4 3 4 4 0 5 1 5 2 5 3 5 4 5 5 0 6 1 6 2 6 3 6 4 6 5 6 6 0 7 1 7 2 7 3 7 4 7 5 7 6 7 7 0 8 1 8 2 8 3 8 4 8 5 8 6 8 7 8 8 0 0 1 1 0 2 1 2 2 0 3 1 3 2 3 3 0 4 1 4 2 4 3 4 4 0 5 1 5 2 5 3 5 4 5 5 0 6 1 6 2 6 3 6 4 6 5 6 6 0 7 1 7 2 7 3 7 4 7 5 7 6 7 7 0 8 1 8 2 8 3 8 4 8 5 8 6 8 7 8 8 0 0 1 1 0 2 1 2 2 0 3 1 3 2 3 3 0 4 1 4 2 4 3 4 4 0 5 1 5 2 5 3 5 4 5 5 0 6 1 6 2 6 3 6 4 6 5 6 6 0 7 1 7 2 7 3 7 4 7 5 7 6 7 7 0 8 1 8 2 8 3 8 4 8 5 8 6 8 7 8 8 0 0 1 1 0 2 1 2 2 0 3 1 3 2 3 3 0 4 1 4 2 4 3 4 4 0 5 1 5 2 5 3 5 4 5 5 0 6 1 6 2 6 3 6 4 6 5 6 6 0 7 1 7 2 7 3 7 4 7 5 7 6 7 7 0 8 1 8 2 8 3 8 4 8 5 8 6 8 7 8 8 0 0 1 1 0 2 1 2 2 0 3 1 3 2 3 3 0 4 1 4 2 4 3 4 4 0 5 1 5 2 5 3 5 4 5 5 0 6 1 6 2 6 3 6 4 6 5 6 6 0 7 1 7 2 7 3 7 4 7 5 7 6 7 7 0 8 1 8 2 8 3 8 4 8 5 8 6 8 7 8 8 0 0 1 1 0 2 1 2 2 0 3 1 3 2 3 3 0 4 1 4 2 4 3 4 4 0 5 1 5 2 5 3 5 4 5 5 0 6 1 6 2 6 3 6 4 6 5 6 6 0 7 1 7 2 7 3 7 4 7 5 7 6 7 7 0 8 1 8 2 8 3 8 4 8 5 8 6 8 7 8 8 0 0 1 1 0 2 1 2 2 0 3 1 3 2 3 3 0 4 1 4 2 4 3 4 4 0 5 1 5 2 5 3 5 4 5 5 0 6 1 6 2 6 3 6 4 6 5 6 6 0 7 1 7 2 7 3 7 4 7 5 7 6 7 7 0 8 1 8 2 8 3 8 4 8 5 8 6 8 7 8 8 0 0 1 1 0 2 1 2 2 0 3 1 3 2 3 3 0 4 1 4 2 4 3 4 4 0 5 1 5 2 5 3 5 4 5 5 0 6 1 6 2 6 3 6 4 6 5 6 6 0 7 1 7 2 7 3 7 4 7 5 7 6 7 7 0 8 1 8 2 8 3 8 4 8 5 8 6 8 7 8 8 0 0 1 1 0 2 1 2 2 0 3 1 3 2 3 3 0 4 1 4 2 4 3 4 4 0 5 1 5 2 5 3 5 4 5 5 0 6 1 6 2 6 3 6 4 6 5 6 6 0 7 1 7 2 7 3 7 4 7 5 7 6 7 7 0 8 1 8 2 8 3 8 4 8 5 8 6 8 7 8 8 0 0 1 1 0 2 1 2 2 0 3 1 3 2 3 3 0 4 1 4 2 4 3 4 4 0 5 1 5 2 5 3 5 4 5 5 0 6 1 6 2 6 3 6 4 6 5 6 6 0 7 1 7 2 7 3 7 4 7 5 7 6 7 7 0 8 1 8 2 8 3 8 4 8 5 8 6 8 7 8 8 o4 : RingMap ringP8 <--- ringP14 i5 : time degreeMap phi -- used 0.0390373 seconds o5 = 1 i6 : -- Compose phi:P^8--->P^14 with a linear projection P^14--->P^8 from a general subspace of P^14 -- of dimension 5 (so that the composition phi':P^8--->P^8 must have degree equal to deg(G(1,5))=14) phi'=phi*map(ringP14,ringP8,for i to 8 list random(1,ringP14)) 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 o6 = map (ringP8, ringP8, {- 780x - 506x x + 1537x - 132x x - 928x x + 386x - 102x x + 422x x + 725x x - 1073x - 905x x - 830x x + 1500x x + 276x x + 1533x - 653x x + 1558x x + 939x x - 1432x x + 462x x - 329x - 92x x + 661x x - 1298x x - 684x x + 70x x - 715x x + 1093x + 581x x + 329x x + 454x x - 911x x - 84x x - 1452x x - 809x x + 1202x + 1353x x + 1503x x + 482x x + 893x x - 643x x + 598x x + 110x x + 1064x x - 472x , - 522x - 583x x + 1339x + 1535x x - 1317x x + 1113x - 169x x + 1440x x - 1657x x + 721x + 40x x - 1576x x - 367x x + 257x x - 1454x + 1612x x + 1529x x - 1068x x + 560x x - 1441x x + 608x - 92x x - 1006x x + 285x x + 102x x - 397x x + 66x x - 643x - 38x x + 1380x x + 1069x x - 426x x + 1147x x + 982x x + 10x x - 662x + 16x x + 1561x x + 1597x x + 512x x + 1288x x - 1253x x + 1317x x + 1481x x - 354x , - 640x - 1551x x + 469x + 1482x x - 1593x x - 986x + 471x x + 612x x + 1228x x + 1156x - 731x x + 1503x x - 628x x + 674x x - 799x + 1137x x + 844x x + 589x x - 666x x + 829x x - 1024x - 170x x + 450x x + 1497x x + 1204x x - 907x x + 1621x x - 417x + 1297x x + 1444x x + 4x x + 398x x + 996x x - 1031x x + 239x x + 303x + 1215x x - 83x x + 1571x x - 1543x x - 925x x - 694x x + 151x x - 520x x + 880x , - 1210x - 222x x + 185x + 245x x + 1059x x - 322x + 238x x + 962x x + 1260x x - 1581x + 50x x + 1352x x - 1465x x + 1555x x + 1333x + 1362x x + 1365x x + 1168x x - 1401x x + 149x x - 652x + 1378x x - 557x x - 112x x + 26x x + 315x x + 111x x + 1592x - 283x x - 1454x x + 907x x + 212x x + 400x x + 1049x x - 882x x - 1429x - 183x x + 1571x x - 1286x x - 1179x x + 1319x x + 240x x - 1100x x + 1500x x - 348x , 1051x - 1325x x + 1354x - 346x x - 1532x x - 466x + 163x x - 659x x - 291x x + 966x + 789x x + 393x x + 403x x - 1199x x - 570x - 93x x - 492x x - 418x x + 713x x - 1323x x - 1384x - 830x x - 54x x - 306x x + 709x x + 421x x - 954x x - 299x + 1053x x - 1080x x + 686x x + 170x x - 1272x x - 1661x x + 1235x x + 1553x - 1454x x - 1411x x - 1195x x - 962x x + 737x x - 390x x + 957x x + 1538x x + 1234x , - 509x + 9x x - 1563x - 710x x - 642x x + 541x + 220x x - 1214x x - 16x x + 1008x - 1088x x + 755x x - 886x x - 1433x x + 1154x + 1627x x - 1547x x - 951x x + 866x x + 163x x - 1142x - 668x x + 1361x x + 1324x x - 490x x + 282x x - 1133x x - 612x + 805x x - 126x x + 1296x x - 973x x + 1271x x - 1646x x + 844x x + 1073x - 1452x x - 1112x x - 141x x + 176x x - 1579x x - 78x x + 848x x - 1365x x + 711x , x + 1543x x - 1076x + 493x x - 526x x + 868x - 582x x - 996x x + 206x x - 419x + 1258x x - 391x x + 1002x x - 1539x x + 931x - 1504x x + 810x x + 324x x + 1356x x + 313x x + 772x + 299x x + 1186x x + 718x x + 407x x - 64x x - 828x x - 1393x + 94x x - 290x x - 766x x + 950x x - 640x x + 265x x - 1640x x - 1403x - 126x x + 891x x - 1519x x - 927x x - 1335x x - 1448x x - x x - 1103x x - 1152x , 821x + 558x x - 1174x - 168x x + 986x x + 790x + 549x x + 817x x + 1396x x + 695x + 1211x x + 878x x - 1061x x - 1244x x - 880x + 1409x x - 567x x + 1240x x + 1126x x - 1262x x + 490x + 1553x x + 1276x x + 805x x + 576x x - 1076x x + 1617x x - 495x - 750x x - 277x x + 544x x + 1479x x - 784x x - 64x x - 1203x x + 405x + 1013x x + 604x x + 1301x x + 1003x x + 235x x + 696x x + 939x x - 714x x - 879x , - 1452x + 727x x - 1159x + 449x x - 1169x x + 732x + 575x x - 600x x + 924x x - 837x + 1298x x - 860x x + 1010x x + 774x x + 319x + 1087x x - 1120x x + 1439x x + 1175x x - 1648x x + 985x - 1317x x - 878x x + 399x x - 1339x x + 70x x - 463x x + 470x - 628x x - 907x x + 748x x + 98x x + 1150x x + 1140x x + 1308x x + 621x + 369x x - 991x x - 1186x x + 61x x - 907x x - 681x x - 1528x x + 717x x + 854x }) 0 0 1 1 0 2 1 2 2 0 3 1 3 2 3 3 0 4 1 4 2 4 3 4 4 0 5 1 5 2 5 3 5 4 5 5 0 6 1 6 2 6 3 6 4 6 5 6 6 0 7 1 7 2 7 3 7 4 7 5 7 6 7 7 0 8 1 8 2 8 3 8 4 8 5 8 6 8 7 8 8 0 0 1 1 0 2 1 2 2 0 3 1 3 2 3 3 0 4 1 4 2 4 3 4 4 0 5 1 5 2 5 3 5 4 5 5 0 6 1 6 2 6 3 6 4 6 5 6 6 0 7 1 7 2 7 3 7 4 7 5 7 6 7 7 0 8 1 8 2 8 3 8 4 8 5 8 6 8 7 8 8 0 0 1 1 0 2 1 2 2 0 3 1 3 2 3 3 0 4 1 4 2 4 3 4 4 0 5 1 5 2 5 3 5 4 5 5 0 6 1 6 2 6 3 6 4 6 5 6 6 0 7 1 7 2 7 3 7 4 7 5 7 6 7 7 0 8 1 8 2 8 3 8 4 8 5 8 6 8 7 8 8 0 0 1 1 0 2 1 2 2 0 3 1 3 2 3 3 0 4 1 4 2 4 3 4 4 0 5 1 5 2 5 3 5 4 5 5 0 6 1 6 2 6 3 6 4 6 5 6 6 0 7 1 7 2 7 3 7 4 7 5 7 6 7 7 0 8 1 8 2 8 3 8 4 8 5 8 6 8 7 8 8 0 0 1 1 0 2 1 2 2 0 3 1 3 2 3 3 0 4 1 4 2 4 3 4 4 0 5 1 5 2 5 3 5 4 5 5 0 6 1 6 2 6 3 6 4 6 5 6 6 0 7 1 7 2 7 3 7 4 7 5 7 6 7 7 0 8 1 8 2 8 3 8 4 8 5 8 6 8 7 8 8 0 0 1 1 0 2 1 2 2 0 3 1 3 2 3 3 0 4 1 4 2 4 3 4 4 0 5 1 5 2 5 3 5 4 5 5 0 6 1 6 2 6 3 6 4 6 5 6 6 0 7 1 7 2 7 3 7 4 7 5 7 6 7 7 0 8 1 8 2 8 3 8 4 8 5 8 6 8 7 8 8 0 0 1 1 0 2 1 2 2 0 3 1 3 2 3 3 0 4 1 4 2 4 3 4 4 0 5 1 5 2 5 3 5 4 5 5 0 6 1 6 2 6 3 6 4 6 5 6 6 0 7 1 7 2 7 3 7 4 7 5 7 6 7 7 0 8 1 8 2 8 3 8 4 8 5 8 6 8 7 8 8 0 0 1 1 0 2 1 2 2 0 3 1 3 2 3 3 0 4 1 4 2 4 3 4 4 0 5 1 5 2 5 3 5 4 5 5 0 6 1 6 2 6 3 6 4 6 5 6 6 0 7 1 7 2 7 3 7 4 7 5 7 6 7 7 0 8 1 8 2 8 3 8 4 8 5 8 6 8 7 8 8 0 0 1 1 0 2 1 2 2 0 3 1 3 2 3 3 0 4 1 4 2 4 3 4 4 0 5 1 5 2 5 3 5 4 5 5 0 6 1 6 2 6 3 6 4 6 5 6 6 0 7 1 7 2 7 3 7 4 7 5 7 6 7 7 0 8 1 8 2 8 3 8 4 8 5 8 6 8 7 8 8 o6 : RingMap ringP8 <--- ringP8 i7 : time degreeMap phi' -- used 0.868663 seconds o7 = 14