1840:
1490:
These two processes projective completion and restriction to an affine subspace are inverse one to the other. Therefore, an affine hypersurface and its projective completion have essentially the same properties, and are often considered as two points-of-view for the same hypersurface.
1474:
1027:
598:
173:
1314:
758:
867:
1162:
1226:
500:. In the case of possibly reducible hypersurfaces, this result may be restated as follows: hypersurfaces are exactly the algebraic sets whose all irreducible components have dimension
342:
1559:
543:
437:, which are the common zeros, if any, of the defining polynomial and its partial derivatives. In particular, a real algebraic hypersurface is not necessarily a manifold.
1087:
416:
763:
is a real hypersurface without any real point, which is defined over the rational numbers. It has no rational point, but has many points that are rational over the
663:
1322:
891:
1664:
83:
A hypersurface in a (Euclidean, affine, or projective) space of dimension two is a plane curve. In a space of dimension three, it is a surface.
1656:
1771:
363:. It may depend on the authors or the context whether a reducible polynomial defines a hypersurface. For avoiding ambiguity, the term
1495:
556:
434:
92:
1637:
1729:
2068:
1097:, and the points of the projective hypersurface that belong to this affine space form an affine hypersurface of equation
1874:
1824:
1242:
695:
475:
258:
179:
523:
coefficients. In this case the algebraically closed field over which the points are defined is generally the field
804:
1100:
1764:
1167:
423:
283:
1233:
1714:
1564:
in the affine space of dimension three has a unique singular point, which is at infinity, in the direction
449:
2063:
1859:
1709:
1498:, while its projective completion has singular points. In this case, one says that the affine surface is
616:
2078:
1757:
1512:
1794:
1704:
526:
2002:
1997:
1977:
445:
Hypersurfaces have some specific properties that are not shared with other algebraic varieties.
352:
73:
492:. This is the geometric interpretation of the fact that, in a polynomial ring over a field, the
370:
As for algebraic varieties, the coefficients of the defining polynomial may belong to any fixed
1987:
1982:
1962:
1740:
Visualization of hypersurfaces and multivariable (objective) functions by partial globalization
1090:
1042:
799:
468:
464:
356:
1625:
1992:
1972:
1967:
1059:
36:
471:) define the same hypersurface, then one is the product of the other by a nonzero constant.
391:
641:
493:
456:
if and only if the defining polynomial of the hypersurface has a power that belongs to the
8:
1869:
1864:
1721:
1469:{\displaystyle P(x_{0},x_{1},\ldots ,x_{n})=x_{0}^{d}p(x_{1}/x_{0},\ldots ,x_{n}/x_{0}),}
457:
371:
1678:
Lima, Elon L. (1988). "The Jordan-Brouwer separation theorem for smooth hypersurfaces".
2043:
1884:
1839:
1733:
1604:
1589:
378:
359:. When this is not the case, the hypersurface is not an algebraic variety, but only an
2073:
1879:
1633:
1599:
1503:
764:
274:
77:
44:
1809:
1691:
1687:
787:
69:
1854:
1799:
1022:{\displaystyle P(cx_{0},cx_{1},\ldots ,cx_{n})=c^{d}P(x_{0},x_{1},\ldots ,x_{n})}
620:
497:
243:
222:
195:
61:
1936:
1921:
1739:
1725:
667:
546:
2057:
1926:
1652:
1584:
453:
360:
246:
239:
1946:
1804:
1594:
1094:
628:
624:
485:
386:
65:
2031:
1814:
520:
199:
32:
16:
Manifold or algebraic variety of dimension n in a space of dimension n+1
2026:
1906:
1053:
604:
of the hypersurface. Often, it is left to the context whether the term
28:
2007:
1916:
1829:
1780:
250:
1931:
1894:
1819:
881:
683:
203:
40:
20:
1941:
1041:
of the hypersurface are the points of the projective space whose
611:
If the coefficients of the defining polynomial belong to a field
1632:. Providence: American Mathematical Society. pp. 143–188.
80:, at least locally (near every point), and sometimes globally.
277:
that may be defined by a single implicit equation of the form
1898:
1749:
460:
generated by the defining polynomials of the algebraic set.
600:
The set of the real points of a real hypersurface is the
593:{\displaystyle \mathbb {R} ^{n}\subset \mathbb {C} ^{n}.}
168:{\displaystyle x_{1}^{2}+x_{2}^{2}+\cdots +x_{n}^{2}-1=0}
1239:. That is, the equation of the projective completion is
519:
is a hypersurface that is defined by a polynomial with
257:
into two connected components; this is related to the
1515:
1494:
However, it may occur that an affine hypersurface is
1325:
1245:
1170:
1164:
Conversely, given an affine hypersurface of equation
1103:
1062:
894:
807:
698:
675:(in the case of the field of rational numbers, "over
644:
559:
553:
of a real hypersurface are the points that belong to
529:
452:, which asserts that a hypersurface contains a given
394:
286:
95:
54:, which is embedded in an ambient space of dimension
770:
1553:
1468:
1308:
1220:
1156:
1081:
1021:
861:
752:
657:
592:
537:
410:
336:
167:
1228:it defines a projective hypersurface, called its
2055:
1665:Proceedings of the American Mathematical Society
264:
1309:{\displaystyle P(x_{0},x_{1},\ldots ,x_{n})=0,}
753:{\displaystyle x_{0}^{2}+\cdots +x_{n}^{2}+1=0}
608:refers to all points or only to the real part.
496:of an ideal is 1 if and only if the ideal is a
474:Hypersurfaces are exactly the subvarieties of
1765:
1626:"Curves and Hypersurfaces in Euclidean Space"
377:, and the points of the hypersurface are the
355:. Generally the polynomial is supposed to be
888:have the same degree, or, equivalently that
862:{\displaystyle P(x_{0},x_{1},\ldots ,x_{n})}
463:A corollary of this theorem is that, if two
76:, the property of being defined by a single
1157:{\displaystyle P(1,x_{1},\ldots ,x_{n})=0.}
510:
1772:
1758:
1093:, the complement of this hyperplane is an
72:. Hypersurfaces share, with surfaces in a
1221:{\displaystyle p(x_{1},\ldots ,x_{n})=0,}
577:
562:
531:
337:{\displaystyle p(x_{1},\ldots ,x_{n})=0,}
27:is a generalization of the concepts of
2056:
216:
1753:
1738:P.A. Simionescu & D. Beal (2004)
1037:is the degree of the polynomial. The
631:), one says that the hypersurface is
178:defines an algebraic hypersurface of
1730:Foundations of Differential Geometry
1677:
188:in the Euclidean space of dimension
1671:
1657:"Orientability of hypersurfaces in
1630:Manifolds and Differential Geometry
1623:
777:projective (algebraic) hypersurface
448:One of the main such properties is
13:
14:
2090:
1680:The American Mathematical Monthly
771:Projective algebraic hypersurface
259:Jordan–Brouwer separation theorem
1838:
1232:, whose equation is obtained by
638:, and the points that belong to
1554:{\displaystyle x^{2}+y^{2}-1=0}
1692:10.1080/00029890.1988.11971963
1646:
1617:
1460:
1398:
1374:
1329:
1294:
1249:
1206:
1174:
1145:
1107:
1016:
971:
952:
898:
856:
811:
424:algebraically closed extension
322:
290:
265:Affine algebraic hypersurface
194:. This hypersurface is also a
1:
1779:
1610:
440:
538:{\displaystyle \mathbb {C} }
7:
1710:Encyclopedia of Mathematics
1578:
682:For example, the imaginary
238:, a smooth hypersurface is
10:
2095:
2069:Multi-dimensional geometry
876:indeterminates. As usual,
86:For example, the equation
2040:
2019:
1955:
1893:
1847:
1836:
1787:
679:" is generally omitted).
450:Hilbert's Nullstellensatz
249:smooth hypersurface is a
221:A hypersurface that is a
689:defined by the equation
619:(typically the field of
511:Real and rational points
433:A hypersurface may have
365:irreducible hypersurface
1082:{\displaystyle x_{0}=0}
469:square-free polynomials
467:(or more generally two
465:irreducible polynomials
353:multivariate polynomial
74:three-dimensional space
1555:
1470:
1310:
1222:
1158:
1091:hyperplane at infinity
1083:
1043:projective coordinates
1023:
878:homogeneous polynomial
863:
800:homogeneous polynomial
754:
659:
594:
539:
412:
411:{\displaystyle K^{n},}
338:
271:algebraic hypersurface
169:
39:. A hypersurface is a
1624:Lee, Jeffrey (2009).
1556:
1471:
1311:
1230:projective completion
1223:
1159:
1084:
1024:
864:
755:
660:
658:{\displaystyle k^{n}}
595:
540:
413:
339:
170:
1956:Dimensions by number
1746:20(10):665–81.
1513:
1500:singular at infinity
1323:
1243:
1168:
1101:
1060:
892:
805:
696:
642:
617:algebraically closed
557:
527:
392:
284:
93:
1744:The Visual Computer
1722:Shoshichi Kobayashi
1502:. For example, the
1394:
1052:If one chooses the
1029:for every constant
737:
713:
227:smooth hypersurface
217:Smooth hypersurface
198:, and is called a
152:
128:
110:
2064:Algebraic geometry
1885:Degrees of freedom
1788:Dimensional spaces
1734:Wiley Interscience
1605:Polar hypersurface
1590:Coble hypersurface
1551:
1466:
1380:
1306:
1218:
1154:
1079:
1019:
859:
765:Gaussian rationals
750:
723:
699:
655:
590:
535:
408:
334:
165:
138:
114:
96:
2051:
2050:
1860:Lebesgue covering
1825:Algebraic variety
1639:978-0-8218-4815-9
1600:Null hypersurface
1504:circular cylinder
1483:is the degree of
517:real hypersurface
275:algebraic variety
78:implicit equation
45:algebraic variety
2086:
2079:Dimension theory
1848:Other dimensions
1842:
1810:Projective space
1774:
1767:
1760:
1751:
1750:
1718:
1696:
1695:
1675:
1669:
1650:
1644:
1643:
1621:
1574:
1560:
1558:
1557:
1552:
1538:
1537:
1525:
1524:
1486:
1482:
1475:
1473:
1472:
1467:
1459:
1458:
1449:
1444:
1443:
1425:
1424:
1415:
1410:
1409:
1393:
1388:
1373:
1372:
1354:
1353:
1341:
1340:
1315:
1313:
1312:
1307:
1293:
1292:
1274:
1273:
1261:
1260:
1238:
1227:
1225:
1224:
1219:
1205:
1204:
1186:
1185:
1163:
1161:
1160:
1155:
1144:
1143:
1125:
1124:
1088:
1086:
1085:
1080:
1072:
1071:
1048:
1036:
1032:
1028:
1026:
1025:
1020:
1015:
1014:
996:
995:
983:
982:
967:
966:
951:
950:
929:
928:
913:
912:
887:
875:
868:
866:
865:
860:
855:
854:
836:
835:
823:
822:
798:is defined by a
797:
793:
788:projective space
785:
759:
757:
756:
751:
736:
731:
712:
707:
686:
678:
674:
664:
662:
661:
656:
654:
653:
637:
621:rational numbers
614:
599:
597:
596:
591:
586:
585:
580:
571:
570:
565:
544:
542:
541:
536:
534:
506:
491:
488:of dimension of
483:
429:
421:
417:
415:
414:
409:
404:
403:
384:
376:
367:is often used.
350:
343:
341:
340:
335:
321:
320:
302:
301:
253:, and separates
237:
210:
193:
187:
174:
172:
171:
166:
151:
146:
127:
122:
109:
104:
70:projective space
59:
53:
2094:
2093:
2089:
2088:
2087:
2085:
2084:
2083:
2054:
2053:
2052:
2047:
2036:
2015:
1951:
1889:
1843:
1834:
1800:Euclidean space
1783:
1778:
1703:
1700:
1699:
1676:
1672:
1651:
1647:
1640:
1622:
1618:
1613:
1581:
1565:
1533:
1529:
1520:
1516:
1514:
1511:
1510:
1484:
1480:
1454:
1450:
1445:
1439:
1435:
1420:
1416:
1411:
1405:
1401:
1389:
1384:
1368:
1364:
1349:
1345:
1336:
1332:
1324:
1321:
1320:
1288:
1284:
1269:
1265:
1256:
1252:
1244:
1241:
1240:
1236:
1200:
1196:
1181:
1177:
1169:
1166:
1165:
1139:
1135:
1120:
1116:
1102:
1099:
1098:
1067:
1063:
1061:
1058:
1057:
1046:
1034:
1030:
1010:
1006:
991:
987:
978:
974:
962:
958:
946:
942:
924:
920:
908:
904:
893:
890:
889:
885:
880:means that all
870:
850:
846:
831:
827:
818:
814:
806:
803:
802:
795:
791:
780:
773:
732:
727:
708:
703:
697:
694:
693:
684:
676:
672:
649:
645:
643:
640:
639:
635:
612:
581:
576:
575:
566:
561:
560:
558:
555:
554:
547:complex numbers
530:
528:
525:
524:
513:
501:
498:principal ideal
489:
478:
443:
427:
419:
399:
395:
393:
390:
389:
382:
374:
348:
316:
312:
297:
293:
285:
282:
281:
267:
233:
223:smooth manifold
219:
204:
196:smooth manifold
189:
182:
147:
142:
123:
118:
105:
100:
94:
91:
90:
62:Euclidean space
55:
48:
17:
12:
11:
5:
2092:
2082:
2081:
2076:
2071:
2066:
2049:
2048:
2041:
2038:
2037:
2035:
2034:
2029:
2023:
2021:
2017:
2016:
2014:
2013:
2005:
2000:
1995:
1990:
1985:
1980:
1975:
1970:
1965:
1959:
1957:
1953:
1952:
1950:
1949:
1944:
1939:
1937:Cross-polytope
1934:
1929:
1924:
1922:Hyperrectangle
1919:
1914:
1909:
1903:
1901:
1891:
1890:
1888:
1887:
1882:
1877:
1872:
1867:
1862:
1857:
1851:
1849:
1845:
1844:
1837:
1835:
1833:
1832:
1827:
1822:
1817:
1812:
1807:
1802:
1797:
1791:
1789:
1785:
1784:
1777:
1776:
1769:
1762:
1754:
1748:
1747:
1736:
1726:Katsumi Nomizu
1719:
1705:"Hypersurface"
1698:
1697:
1670:
1645:
1638:
1615:
1614:
1612:
1609:
1608:
1607:
1602:
1597:
1592:
1587:
1580:
1577:
1562:
1561:
1550:
1547:
1544:
1541:
1536:
1532:
1528:
1523:
1519:
1501:
1477:
1476:
1465:
1462:
1457:
1453:
1448:
1442:
1438:
1434:
1431:
1428:
1423:
1419:
1414:
1408:
1404:
1400:
1397:
1392:
1387:
1383:
1379:
1376:
1371:
1367:
1363:
1360:
1357:
1352:
1348:
1344:
1339:
1335:
1331:
1328:
1305:
1302:
1299:
1296:
1291:
1287:
1283:
1280:
1277:
1272:
1268:
1264:
1259:
1255:
1251:
1248:
1231:
1217:
1214:
1211:
1208:
1203:
1199:
1195:
1192:
1189:
1184:
1180:
1176:
1173:
1153:
1150:
1147:
1142:
1138:
1134:
1131:
1128:
1123:
1119:
1115:
1112:
1109:
1106:
1078:
1075:
1070:
1066:
1040:
1018:
1013:
1009:
1005:
1002:
999:
994:
990:
986:
981:
977:
973:
970:
965:
961:
957:
954:
949:
945:
941:
938:
935:
932:
927:
923:
919:
916:
911:
907:
903:
900:
897:
879:
858:
853:
849:
845:
842:
839:
834:
830:
826:
821:
817:
813:
810:
778:
772:
769:
761:
760:
749:
746:
743:
740:
735:
730:
726:
722:
719:
716:
711:
706:
702:
652:
648:
589:
584:
579:
574:
569:
564:
533:
512:
509:
442:
439:
407:
402:
398:
345:
344:
333:
330:
327:
324:
319:
315:
311:
308:
305:
300:
296:
292:
289:
266:
263:
218:
215:
176:
175:
164:
161:
158:
155:
150:
145:
141:
137:
134:
131:
126:
121:
117:
113:
108:
103:
99:
60:, generally a
15:
9:
6:
4:
3:
2:
2091:
2080:
2077:
2075:
2072:
2070:
2067:
2065:
2062:
2061:
2059:
2046:
2045:
2039:
2033:
2030:
2028:
2025:
2024:
2022:
2018:
2012:
2010:
2006:
2004:
2001:
1999:
1996:
1994:
1991:
1989:
1986:
1984:
1981:
1979:
1976:
1974:
1971:
1969:
1966:
1964:
1961:
1960:
1958:
1954:
1948:
1945:
1943:
1940:
1938:
1935:
1933:
1930:
1928:
1927:Demihypercube
1925:
1923:
1920:
1918:
1915:
1913:
1910:
1908:
1905:
1904:
1902:
1900:
1896:
1892:
1886:
1883:
1881:
1878:
1876:
1873:
1871:
1868:
1866:
1863:
1861:
1858:
1856:
1853:
1852:
1850:
1846:
1841:
1831:
1828:
1826:
1823:
1821:
1818:
1816:
1813:
1811:
1808:
1806:
1803:
1801:
1798:
1796:
1793:
1792:
1790:
1786:
1782:
1775:
1770:
1768:
1763:
1761:
1756:
1755:
1752:
1745:
1741:
1737:
1735:
1731:
1727:
1723:
1720:
1716:
1712:
1711:
1706:
1702:
1701:
1693:
1689:
1685:
1681:
1674:
1668:22(1): 301,2
1667:
1666:
1661:
1660:
1654:
1653:Hans Samelson
1649:
1641:
1635:
1631:
1627:
1620:
1616:
1606:
1603:
1601:
1598:
1596:
1593:
1591:
1588:
1586:
1585:Affine sphere
1583:
1582:
1576:
1572:
1568:
1548:
1545:
1542:
1539:
1534:
1530:
1526:
1521:
1517:
1509:
1508:
1507:
1506:of equation
1505:
1499:
1497:
1492:
1488:
1463:
1455:
1451:
1446:
1440:
1436:
1432:
1429:
1426:
1421:
1417:
1412:
1406:
1402:
1395:
1390:
1385:
1381:
1377:
1369:
1365:
1361:
1358:
1355:
1350:
1346:
1342:
1337:
1333:
1326:
1319:
1318:
1317:
1303:
1300:
1297:
1289:
1285:
1281:
1278:
1275:
1270:
1266:
1262:
1257:
1253:
1246:
1235:
1229:
1215:
1212:
1209:
1201:
1197:
1193:
1190:
1187:
1182:
1178:
1171:
1151:
1148:
1140:
1136:
1132:
1129:
1126:
1121:
1117:
1113:
1110:
1104:
1096:
1092:
1076:
1073:
1068:
1064:
1055:
1050:
1045:are zeros of
1044:
1038:
1011:
1007:
1003:
1000:
997:
992:
988:
984:
979:
975:
968:
963:
959:
955:
947:
943:
939:
936:
933:
930:
925:
921:
917:
914:
909:
905:
901:
895:
883:
877:
873:
851:
847:
843:
840:
837:
832:
828:
824:
819:
815:
808:
801:
794:over a field
790:of dimension
789:
783:
779:of dimension
776:
768:
766:
747:
744:
741:
738:
733:
728:
724:
720:
717:
714:
709:
704:
700:
692:
691:
690:
688:
680:
670:
669:
650:
646:
634:
630:
626:
622:
618:
609:
607:
603:
587:
582:
572:
567:
552:
548:
522:
518:
508:
504:
499:
495:
487:
481:
477:
472:
470:
466:
461:
459:
455:
454:algebraic set
451:
446:
438:
436:
435:singularities
431:
425:
405:
400:
396:
388:
380:
373:
368:
366:
362:
361:algebraic set
358:
354:
331:
328:
325:
317:
313:
309:
306:
303:
298:
294:
287:
280:
279:
278:
276:
272:
262:
260:
256:
252:
248:
245:
241:
236:
230:
228:
224:
214:
212:
208:
201:
197:
192:
185:
181:
162:
159:
156:
153:
148:
143:
139:
135:
132:
129:
124:
119:
115:
111:
106:
101:
97:
89:
88:
87:
84:
81:
79:
75:
71:
67:
63:
58:
51:
47:of dimension
46:
42:
38:
34:
30:
26:
22:
2042:
2008:
1947:Hyperpyramid
1912:Hypersurface
1911:
1805:Affine space
1795:Vector space
1743:
1708:
1686:(1): 39–42.
1683:
1679:
1673:
1663:
1658:
1648:
1629:
1619:
1595:Dwork family
1570:
1566:
1563:
1493:
1489:
1478:
1234:homogenizing
1095:affine space
1056:of equation
1051:
871:
781:
774:
762:
681:
666:
633:defined over
632:
629:number field
625:finite field
615:that is not
610:
606:hypersurface
605:
601:
550:
516:
514:
502:
486:affine space
479:
473:
462:
447:
444:
432:
387:affine space
369:
364:
346:
270:
268:
254:
234:
231:
226:
225:is called a
220:
206:
190:
183:
177:
85:
82:
66:affine space
56:
49:
25:hypersurface
24:
18:
2032:Codimension
2011:-dimensions
1932:Hypersphere
1815:Free module
1496:nonsingular
551:real points
357:irreducible
200:hypersphere
33:plane curve
2058:Categories
2027:Hyperspace
1907:Hyperplane
1611:References
1054:hyperplane
441:Properties
240:orientable
29:hyperplane
1917:Hypercube
1895:Polytopes
1875:Minkowski
1870:Hausdorff
1865:Inductive
1830:Spacetime
1781:Dimension
1715:EMS Press
1540:−
1430:…
1359:…
1279:…
1191:…
1130:…
1001:…
934:…
882:monomials
841:…
718:⋯
602:real part
573:⊂
476:dimension
307:…
251:level set
244:connected
180:dimension
154:−
133:⋯
2074:Surfaces
2044:Category
2020:See also
1820:Manifold
1732:Vol II,
1728:(1969),
1579:See also
1033:, where
668:rational
242:. Every
41:manifold
21:geometry
1942:Simplex
1880:Fractal
1717:, 2001
1655:(1969)
687:-sphere
385:in the
247:compact
211:-sphere
37:surface
1899:shapes
1636:
1479:where
1316:with
1039:points
549:. The
494:height
484:of an
422:is an
418:where
347:where
273:is an
202:or an
43:or an
35:, and
2003:Eight
1998:Seven
1978:Three
1855:Krull
1569:= 0,
786:in a
671:over
627:or a
458:ideal
379:zeros
372:field
351:is a
68:or a
64:, an
1988:Five
1983:Four
1963:Zero
1897:and
1724:and
1634:ISBN
665:are
623:, a
521:real
209:– 1)
23:, a
1993:Six
1973:Two
1968:One
1688:doi
1662:",
1573:= 0
1089:as
884:of
874:+ 1
869:in
784:– 1
545:of
505:– 1
482:– 1
426:of
381:of
269:An
232:In
186:− 1
52:− 1
19:In
2060::
1742:,
1713:,
1707:,
1684:95
1682:.
1628:.
1575:.
1487:.
1152:0.
1049:.
775:A
767:.
515:A
507:.
430:.
261:.
229:.
213:.
31:,
2009:n
1773:e
1766:t
1759:v
1694:.
1690::
1659:R
1642:.
1571:y
1567:x
1549:0
1546:=
1543:1
1535:2
1531:y
1527:+
1522:2
1518:x
1485:P
1481:d
1464:,
1461:)
1456:0
1452:x
1447:/
1441:n
1437:x
1433:,
1427:,
1422:0
1418:x
1413:/
1407:1
1403:x
1399:(
1396:p
1391:d
1386:0
1382:x
1378:=
1375:)
1370:n
1366:x
1362:,
1356:,
1351:1
1347:x
1343:,
1338:0
1334:x
1330:(
1327:P
1304:,
1301:0
1298:=
1295:)
1290:n
1286:x
1282:,
1276:,
1271:1
1267:x
1263:,
1258:0
1254:x
1250:(
1247:P
1237:p
1216:,
1213:0
1210:=
1207:)
1202:n
1198:x
1194:,
1188:,
1183:1
1179:x
1175:(
1172:p
1149:=
1146:)
1141:n
1137:x
1133:,
1127:,
1122:1
1118:x
1114:,
1111:1
1108:(
1105:P
1077:0
1074:=
1069:0
1065:x
1047:P
1035:d
1031:c
1017:)
1012:n
1008:x
1004:,
998:,
993:1
989:x
985:,
980:0
976:x
972:(
969:P
964:d
960:c
956:=
953:)
948:n
944:x
940:c
937:,
931:,
926:1
922:x
918:c
915:,
910:0
906:x
902:c
899:(
896:P
886:P
872:n
857:)
852:n
848:x
844:,
838:,
833:1
829:x
825:,
820:0
816:x
812:(
809:P
796:k
792:n
782:n
748:0
745:=
742:1
739:+
734:2
729:n
725:x
721:+
715:+
710:2
705:0
701:x
685:n
677:k
673:k
651:n
647:k
636:k
613:k
588:.
583:n
578:C
568:n
563:R
532:C
503:n
490:n
480:n
428:k
420:K
406:,
401:n
397:K
383:p
375:k
349:p
332:,
329:0
326:=
323:)
318:n
314:x
310:,
304:,
299:1
295:x
291:(
288:p
255:R
235:R
207:n
205:(
191:n
184:n
163:0
160:=
157:1
149:2
144:n
140:x
136:+
130:+
125:2
120:2
116:x
112:+
107:2
102:1
98:x
57:n
50:n
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