36:
2155:
1543:
649:
461:
1080:
1839:
1242:
1969:
1291:
1673:
1377:
182:
1148:
241:
535:
675:
516:
484:
1331:
807:
896:
858:
349:
753:
827:
923:
1684:
1160:
1086:
243:
is not homogeneous, because the sum of exponents does not match from term to term. The function defined by a homogeneous polynomial is always a
2206:
2122:
2000:
1857:
684:
Any nonzero polynomial may be decomposed, in a unique way, as a sum of homogeneous polynomials of different degrees, which are called the
2386:
1250:
2025:
2015:
79:
57:
2159:
1566:
2437:
1538:{\displaystyle {^{h}\!P}(x_{0},x_{1},\dots ,x_{n})=x_{0}^{d}P\left({\frac {x_{1}}{x_{0}}},\dots ,{\frac {x_{n}}{x_{0}}}\right),}
184:
is a homogeneous polynomial of degree 5, in two variables; the sum of the exponents in each term is always 5. The polynomial
2442:
2381:
2365:
1985:
1980:
317:
2447:
2401:
2044:
However, as some authors do not make a clear distinction between a polynomial and its associated function, the terms
2070:, which are defined for every vector space. "Linear functional" is rarely used for finite-dimensional vector spaces.
2246:
2199:
2391:
115:
50:
44:
2330:
333:
2355:
2350:
2228:
1553:
1104:
275:
259:
109:
61:
644:{\displaystyle P(x_{1},\ldots ,x_{n})=0\quad \Rightarrow \quad P(\lambda x_{1},\ldots ,\lambda x_{n})=0,}
187:
2340:
2320:
2406:
2325:
2192:
2080:
915:
776:
329:
244:
2219:
861:
756:
487:
312:
Homogeneous polynomials are ubiquitous in mathematics and physics. They play a fundamental role in
282:
2263:
2258:
760:
678:
313:
302:
286:
2396:
657:
2241:
2167:
2118:
2066:
1294:
501:
469:
456:{\displaystyle P(\lambda x_{1},\ldots ,\lambda x_{n})=\lambda ^{d}\,P(x_{1},\ldots ,x_{n})\,,}
2064:
are defined only for finite-dimensional vector space, and have thus to be distinguished from
906:
variables (that is the maximal number of nonzero terms in a homogeneous polynomial of degree
2416:
2294:
2287:
2282:
2112:
2020:
2005:
1306:
1151:
782:
2170:
874:
836:
2304:
2299:
2251:
2236:
2010:
692:
2108:
697:
2275:
2270:
2184:
1995:
1075:{\displaystyle {\binom {d+n-1}{n-1}}={\binom {d+n-1}{d}}={\frac {(d+n-1)!}{d!(n-1)!}}.}
812:
294:
1834:{\displaystyle ^{h}\!P(x_{0},x_{1},x_{2},x_{3})=x_{3}^{3}+x_{0}x_{1}x_{2}+7x_{0}^{3}.}
2431:
1990:
2117:. Graduate Texts in Mathematics. Vol. 185 (2nd ed.). Springer. p. 2.
289:
of the coefficients, usually called a constant or a scalar. A form of degree 1 is a
2411:
2104:
772:
271:
17:
1844:
A homogenized polynomial can be dehomogenized by setting the additional variable
868:
491:
320:
is defined as the set of the common zeros of a set of homogeneous polynomials.
306:
290:
93:
281:
A polynomial of degree 0 is always homogeneous; it is simply an element of the
274:, which may be expressed as a homogeneous function of the coordinates over any
2215:
830:
105:
2345:
2175:
1237:{\displaystyle dP=\sum _{i=1}^{n}x_{i}{\frac {\partial P}{\partial x_{i}}},}
2154:
101:
2335:
298:
1964:{\displaystyle P(x_{1},\dots ,x_{n})={^{h}\!P}(1,x_{1},\dots ,x_{n}).}
2079:
Homogeneous polynomials in physics often appear as a consequence of
498:. Conversely, if the above relation is true for infinitely many
2188:
1286:{\displaystyle \textstyle {\frac {\partial P}{\partial x_{i}}}}
2083:, where measured quantities must match in real-world problems.
29:
1668:{\displaystyle P(x_{1},x_{2},x_{3})=x_{3}^{3}+x_{1}x_{2}+7,}
1360:) can be homogenized by introducing an additional variable
1367:
and defining the homogeneous polynomial sometimes denoted
27:
Polynomial whose nonzero terms all have the same degree
1254:
1860:
1687:
1569:
1380:
1309:
1253:
1163:
1107:
926:
877:
839:
815:
785:
700:
660:
538:
504:
472:
352:
190:
118:
677:
This property is fundamental in the definition of a
2374:
2313:
2226:
2137:
1963:
1833:
1667:
1537:
1325:
1285:
1236:
1142:
1074:
890:
852:
821:
801:
747:
669:
643:
510:
478:
455:
235:
176:
1912:
1697:
1391:
1004:
977:
965:
930:
898:is the number of different monomials of degree
2237:Zero polynomial (degree undefined or −1 or −∞)
2200:
518:then the polynomial is homogeneous of degree
8:
2207:
2193:
2185:
1087:Euler's identity for homogeneous functions
809:The above unique decomposition means that
1949:
1930:
1906:
1902:
1890:
1871:
1859:
1822:
1817:
1801:
1791:
1781:
1768:
1763:
1747:
1734:
1721:
1708:
1691:
1686:
1650:
1640:
1627:
1622:
1606:
1593:
1580:
1568:
1519:
1509:
1503:
1486:
1476:
1470:
1456:
1451:
1435:
1416:
1403:
1385:
1381:
1379:
1314:
1308:
1273:
1255:
1252:
1222:
1204:
1198:
1188:
1177:
1162:
1131:
1112:
1106:
1013:
1003:
976:
974:
964:
929:
927:
925:
882:
876:
844:
838:
814:
790:
784:
736:
717:
699:
659:
623:
601:
568:
549:
537:
503:
471:
449:
440:
421:
410:
404:
388:
366:
351:
227:
211:
195:
189:
177:{\displaystyle x^{5}+2x^{3}y^{2}+9xy^{4}}
168:
149:
139:
123:
117:
80:Learn how and when to remove this message
767:, the homogeneous polynomials of degree
43:This article includes a list of general
2096:
2052:are sometimes considered as synonymous.
2037:
262:defined by a homogeneous polynomial. A
1095:is a homogeneous polynomial of degree
867:The dimension of the vector space (or
108:whose nonzero terms all have the same
2001:Hilbert series and Hilbert polynomial
7:
1143:{\displaystyle x_{1},\ldots ,x_{n},}
328:A homogeneous polynomial defines a
236:{\displaystyle x^{3}+3x^{2}y+z^{7}}
1266:
1258:
1215:
1207:
981:
934:
49:it lacks sufficient corresponding
25:
2026:Symbol of a differential operator
2016:Polarization of an algebraic form
2153:
1085:Homogeneous polynomial satisfy
270:is also a function defined on a
34:
914:variables). It is equal to the
587:
583:
1955:
1917:
1896:
1864:
1753:
1701:
1612:
1573:
1441:
1396:
1060:
1048:
1034:
1016:
742:
710:
629:
591:
584:
574:
542:
446:
414:
394:
356:
266:is a form in two variables. A
1:
2397:Horner's method of evaluation
2138:Cox, Little & O'Shea 2005
1340:A non-homogeneous polynomial
1986:Quasi-homogeneous polynomial
1981:Multi-homogeneous polynomial
318:projective algebraic variety
2402:Polynomial identity testing
2464:
1150:one has, whichever is the
293:. A form of degree 2 is a
1295:formal partial derivative
670:{\displaystyle \lambda .}
339:is homogeneous of degree
2171:"Homogeneous Polynomial"
2114:Using Algebraic Geometry
511:{\displaystyle \lambda }
479:{\displaystyle \lambda }
332:. This means that, if a
2438:Homogeneous polynomials
2387:Greatest common divisor
2160:Homogeneous polynomials
759:(or, more generally, a
334:multivariate polynomial
64:more precise citations.
2259:Quadratic function (2)
2046:homogeneous polynomial
1965:
1835:
1669:
1539:
1327:
1326:{\displaystyle x_{i}.}
1287:
1238:
1193:
1144:
1101:in the indeterminates
1076:
892:
854:
823:
803:
802:{\displaystyle R_{d}.}
749:
686:homogeneous components
671:
645:
512:
480:
457:
237:
178:
98:homogeneous polynomial
2242:Constant function (0)
1966:
1836:
1670:
1540:
1328:
1288:
1239:
1173:
1154:of the coefficients,
1145:
1077:
893:
891:{\displaystyle R_{d}}
855:
853:{\displaystyle R_{d}}
824:
804:
750:
672:
646:
529:is homogeneous then
513:
481:
458:
309:of a quadratic form.
238:
179:
104:in older texts, is a
2375:Tools and algorithms
2295:Quintic function (5)
2283:Quartic function (4)
2220:polynomial functions
2162:at Wikimedia Commons
2081:dimensional analysis
1858:
1685:
1567:
1378:
1307:
1251:
1161:
1105:
924:
916:binomial coefficient
875:
862:nonnegative integers
837:
813:
783:
779:), commonly denoted
698:
688:of the polynomial.
658:
536:
502:
470:
350:
330:homogeneous function
245:homogeneous function
188:
116:
2443:Multilinear algebra
2305:Septic equation (7)
2300:Sextic equation (6)
2247:Linear function (1)
1827:
1773:
1632:
1461:
748:{\displaystyle R=K}
100:, sometimes called
2448:Algebraic geometry
2271:Cubic function (3)
2264:Quadratic equation
2168:Weisstein, Eric W.
2067:linear functionals
1961:
1831:
1813:
1759:
1665:
1618:
1560:. For example, if
1535:
1447:
1323:
1283:
1282:
1234:
1140:
1072:
888:
850:
819:
799:
745:
679:projective variety
667:
641:
525:In particular, if
508:
476:
453:
314:algebraic geometry
303:Euclidean distance
233:
174:
18:Form (mathematics)
2425:
2424:
2366:Quasi-homogeneous
2158:Media related to
2124:978-0-387-20733-9
1525:
1492:
1280:
1229:
1067:
1002:
963:
822:{\displaystyle R}
90:
89:
82:
16:(Redirected from
2455:
2288:Quartic equation
2209:
2202:
2195:
2186:
2181:
2180:
2157:
2141:
2135:
2129:
2128:
2107:; Little, John;
2101:
2084:
2077:
2071:
2059:
2053:
2042:
2021:Schur polynomial
2006:Multilinear form
1970:
1968:
1967:
1962:
1954:
1953:
1935:
1934:
1916:
1911:
1910:
1895:
1894:
1876:
1875:
1840:
1838:
1837:
1832:
1826:
1821:
1806:
1805:
1796:
1795:
1786:
1785:
1772:
1767:
1752:
1751:
1739:
1738:
1726:
1725:
1713:
1712:
1696:
1695:
1674:
1672:
1671:
1666:
1655:
1654:
1645:
1644:
1631:
1626:
1611:
1610:
1598:
1597:
1585:
1584:
1544:
1542:
1541:
1536:
1531:
1527:
1526:
1524:
1523:
1514:
1513:
1504:
1493:
1491:
1490:
1481:
1480:
1471:
1460:
1455:
1440:
1439:
1421:
1420:
1408:
1407:
1395:
1390:
1389:
1332:
1330:
1329:
1324:
1319:
1318:
1303:with respect to
1302:
1292:
1290:
1289:
1284:
1281:
1279:
1278:
1277:
1264:
1256:
1243:
1241:
1240:
1235:
1230:
1228:
1227:
1226:
1213:
1205:
1203:
1202:
1192:
1187:
1152:commutative ring
1149:
1147:
1146:
1141:
1136:
1135:
1117:
1116:
1100:
1094:
1081:
1079:
1078:
1073:
1068:
1066:
1040:
1014:
1009:
1008:
1007:
998:
980:
970:
969:
968:
962:
951:
933:
897:
895:
894:
889:
887:
886:
859:
857:
856:
851:
849:
848:
828:
826:
825:
820:
808:
806:
805:
800:
795:
794:
754:
752:
751:
746:
741:
740:
722:
721:
676:
674:
673:
668:
650:
648:
647:
642:
628:
627:
606:
605:
573:
572:
554:
553:
517:
515:
514:
509:
485:
483:
482:
477:
462:
460:
459:
454:
445:
444:
426:
425:
409:
408:
393:
392:
371:
370:
242:
240:
239:
234:
232:
231:
216:
215:
200:
199:
183:
181:
180:
175:
173:
172:
154:
153:
144:
143:
128:
127:
85:
78:
74:
71:
65:
60:this article by
51:inline citations
38:
37:
30:
21:
2463:
2462:
2458:
2457:
2456:
2454:
2453:
2452:
2428:
2427:
2426:
2421:
2370:
2309:
2252:Linear equation
2222:
2213:
2166:
2165:
2150:
2145:
2144:
2136:
2132:
2125:
2103:
2102:
2098:
2093:
2088:
2087:
2078:
2074:
2060:
2056:
2043:
2039:
2034:
2011:Multilinear map
1977:
1945:
1926:
1903:
1886:
1867:
1856:
1855:
1850:
1797:
1787:
1777:
1743:
1730:
1717:
1704:
1688:
1683:
1682:
1646:
1636:
1602:
1589:
1576:
1565:
1564:
1515:
1505:
1482:
1472:
1469:
1465:
1431:
1412:
1399:
1382:
1376:
1375:
1366:
1359:
1350:
1338:
1310:
1305:
1304:
1298:
1269:
1265:
1257:
1249:
1248:
1218:
1214:
1206:
1194:
1159:
1158:
1127:
1108:
1103:
1102:
1096:
1090:
1041:
1015:
982:
975:
952:
935:
928:
922:
921:
878:
873:
872:
840:
835:
834:
811:
810:
786:
781:
780:
732:
713:
696:
695:
693:polynomial ring
656:
655:
619:
597:
564:
545:
534:
533:
500:
499:
490:containing the
468:
467:
436:
417:
400:
384:
362:
348:
347:
326:
223:
207:
191:
186:
185:
164:
145:
135:
119:
114:
113:
112:. For example,
86:
75:
69:
66:
56:Please help to
55:
39:
35:
28:
23:
22:
15:
12:
11:
5:
2461:
2459:
2451:
2450:
2445:
2440:
2430:
2429:
2423:
2422:
2420:
2419:
2414:
2409:
2404:
2399:
2394:
2389:
2384:
2378:
2376:
2372:
2371:
2369:
2368:
2363:
2358:
2353:
2348:
2343:
2338:
2333:
2328:
2323:
2317:
2315:
2311:
2310:
2308:
2307:
2302:
2297:
2292:
2291:
2290:
2280:
2279:
2278:
2276:Cubic equation
2268:
2267:
2266:
2256:
2255:
2254:
2244:
2239:
2233:
2231:
2224:
2223:
2214:
2212:
2211:
2204:
2197:
2189:
2183:
2182:
2163:
2149:
2148:External links
2146:
2143:
2142:
2130:
2123:
2095:
2094:
2092:
2089:
2086:
2085:
2072:
2054:
2036:
2035:
2033:
2030:
2029:
2028:
2023:
2018:
2013:
2008:
2003:
1998:
1996:Graded algebra
1993:
1988:
1983:
1976:
1973:
1972:
1971:
1960:
1957:
1952:
1948:
1944:
1941:
1938:
1933:
1929:
1925:
1922:
1919:
1915:
1909:
1905:
1901:
1898:
1893:
1889:
1885:
1882:
1879:
1874:
1870:
1866:
1863:
1848:
1842:
1841:
1830:
1825:
1820:
1816:
1812:
1809:
1804:
1800:
1794:
1790:
1784:
1780:
1776:
1771:
1766:
1762:
1758:
1755:
1750:
1746:
1742:
1737:
1733:
1729:
1724:
1720:
1716:
1711:
1707:
1703:
1700:
1694:
1690:
1676:
1675:
1664:
1661:
1658:
1653:
1649:
1643:
1639:
1635:
1630:
1625:
1621:
1617:
1614:
1609:
1605:
1601:
1596:
1592:
1588:
1583:
1579:
1575:
1572:
1546:
1545:
1534:
1530:
1522:
1518:
1512:
1508:
1502:
1499:
1496:
1489:
1485:
1479:
1475:
1468:
1464:
1459:
1454:
1450:
1446:
1443:
1438:
1434:
1430:
1427:
1424:
1419:
1415:
1411:
1406:
1402:
1398:
1394:
1388:
1384:
1364:
1355:
1348:
1337:
1336:Homogenization
1334:
1322:
1317:
1313:
1276:
1272:
1268:
1263:
1260:
1245:
1244:
1233:
1225:
1221:
1217:
1212:
1209:
1201:
1197:
1191:
1186:
1183:
1180:
1176:
1172:
1169:
1166:
1139:
1134:
1130:
1126:
1123:
1120:
1115:
1111:
1089:. That is, if
1083:
1082:
1071:
1065:
1062:
1059:
1056:
1053:
1050:
1047:
1044:
1039:
1036:
1033:
1030:
1027:
1024:
1021:
1018:
1012:
1006:
1001:
997:
994:
991:
988:
985:
979:
973:
967:
961:
958:
955:
950:
947:
944:
941:
938:
932:
885:
881:
860:(sum over all
847:
843:
818:
798:
793:
789:
744:
739:
735:
731:
728:
725:
720:
716:
712:
709:
706:
703:
666:
663:
652:
651:
640:
637:
634:
631:
626:
622:
618:
615:
612:
609:
604:
600:
596:
593:
590:
586:
582:
579:
576:
571:
567:
563:
560:
557:
552:
548:
544:
541:
507:
475:
464:
463:
452:
448:
443:
439:
435:
432:
429:
424:
420:
416:
413:
407:
403:
399:
396:
391:
387:
383:
380:
377:
374:
369:
365:
361:
358:
355:
325:
322:
295:quadratic form
252:algebraic form
230:
226:
222:
219:
214:
210:
206:
203:
198:
194:
171:
167:
163:
160:
157:
152:
148:
142:
138:
134:
131:
126:
122:
88:
87:
42:
40:
33:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
2460:
2449:
2446:
2444:
2441:
2439:
2436:
2435:
2433:
2418:
2417:Gröbner basis
2415:
2413:
2410:
2408:
2405:
2403:
2400:
2398:
2395:
2393:
2390:
2388:
2385:
2383:
2382:Factorization
2380:
2379:
2377:
2373:
2367:
2364:
2362:
2359:
2357:
2354:
2352:
2349:
2347:
2344:
2342:
2339:
2337:
2334:
2332:
2329:
2327:
2324:
2322:
2319:
2318:
2316:
2314:By properties
2312:
2306:
2303:
2301:
2298:
2296:
2293:
2289:
2286:
2285:
2284:
2281:
2277:
2274:
2273:
2272:
2269:
2265:
2262:
2261:
2260:
2257:
2253:
2250:
2249:
2248:
2245:
2243:
2240:
2238:
2235:
2234:
2232:
2230:
2225:
2221:
2217:
2210:
2205:
2203:
2198:
2196:
2191:
2190:
2187:
2178:
2177:
2172:
2169:
2164:
2161:
2156:
2152:
2151:
2147:
2139:
2134:
2131:
2126:
2120:
2116:
2115:
2110:
2109:O'Shea, Donal
2106:
2105:Cox, David A.
2100:
2097:
2090:
2082:
2076:
2073:
2069:
2068:
2063:
2058:
2055:
2051:
2047:
2041:
2038:
2031:
2027:
2024:
2022:
2019:
2017:
2014:
2012:
2009:
2007:
2004:
2002:
1999:
1997:
1994:
1992:
1991:Diagonal form
1989:
1987:
1984:
1982:
1979:
1978:
1974:
1958:
1950:
1946:
1942:
1939:
1936:
1931:
1927:
1923:
1920:
1913:
1907:
1904:
1899:
1891:
1887:
1883:
1880:
1877:
1872:
1868:
1861:
1854:
1853:
1852:
1851:= 1. That is
1847:
1828:
1823:
1818:
1814:
1810:
1807:
1802:
1798:
1792:
1788:
1782:
1778:
1774:
1769:
1764:
1760:
1756:
1748:
1744:
1740:
1735:
1731:
1727:
1722:
1718:
1714:
1709:
1705:
1698:
1692:
1689:
1681:
1680:
1679:
1662:
1659:
1656:
1651:
1647:
1641:
1637:
1633:
1628:
1623:
1619:
1615:
1607:
1603:
1599:
1594:
1590:
1586:
1581:
1577:
1570:
1563:
1562:
1561:
1559:
1555:
1551:
1532:
1528:
1520:
1516:
1510:
1506:
1500:
1497:
1494:
1487:
1483:
1477:
1473:
1466:
1462:
1457:
1452:
1448:
1444:
1436:
1432:
1428:
1425:
1422:
1417:
1413:
1409:
1404:
1400:
1392:
1386:
1383:
1374:
1373:
1372:
1370:
1363:
1358:
1354:
1347:
1343:
1335:
1333:
1320:
1315:
1311:
1301:
1296:
1274:
1270:
1261:
1231:
1223:
1219:
1210:
1199:
1195:
1189:
1184:
1181:
1178:
1174:
1170:
1167:
1164:
1157:
1156:
1155:
1153:
1137:
1132:
1128:
1124:
1121:
1118:
1113:
1109:
1099:
1093:
1088:
1069:
1063:
1057:
1054:
1051:
1045:
1042:
1037:
1031:
1028:
1025:
1022:
1019:
1010:
999:
995:
992:
989:
986:
983:
971:
959:
956:
953:
948:
945:
942:
939:
936:
920:
919:
918:
917:
913:
909:
905:
901:
883:
879:
870:
865:
863:
845:
841:
832:
816:
796:
791:
787:
778:
774:
770:
766:
762:
758:
737:
733:
729:
726:
723:
718:
714:
707:
704:
701:
694:
689:
687:
682:
680:
664:
661:
638:
635:
632:
624:
620:
616:
613:
610:
607:
602:
598:
594:
588:
580:
577:
569:
565:
561:
558:
555:
550:
546:
539:
532:
531:
530:
528:
523:
521:
505:
497:
493:
489:
473:
450:
441:
437:
433:
430:
427:
422:
418:
411:
405:
401:
397:
389:
385:
381:
378:
375:
372:
367:
363:
359:
353:
346:
345:
344:
342:
338:
335:
331:
323:
321:
319:
315:
310:
308:
304:
300:
296:
292:
288:
284:
279:
277:
273:
269:
265:
261:
257:
253:
248:
246:
228:
224:
220:
217:
212:
208:
204:
201:
196:
192:
169:
165:
161:
158:
155:
150:
146:
140:
136:
132:
129:
124:
120:
111:
107:
103:
99:
95:
84:
81:
73:
63:
59:
53:
52:
46:
41:
32:
31:
19:
2412:Discriminant
2360:
2331:Multivariate
2174:
2140:, p. 35
2133:
2113:
2099:
2075:
2065:
2062:Linear forms
2061:
2057:
2049:
2045:
2040:
1845:
1843:
1677:
1557:
1549:
1547:
1368:
1361:
1356:
1352:
1345:
1341:
1339:
1299:
1293:denotes the
1246:
1097:
1091:
1084:
911:
907:
903:
899:
866:
773:vector space
768:
764:
690:
685:
683:
653:
526:
524:
519:
495:
492:coefficients
465:
340:
336:
327:
311:
280:
272:vector space
267:
263:
255:
254:, or simply
251:
249:
97:
91:
76:
67:
48:
2361:Homogeneous
2356:Square-free
2351:Irreducible
2216:Polynomials
869:free module
307:square root
291:linear form
264:binary form
94:mathematics
62:introducing
2432:Categories
2321:Univariate
2091:References
831:direct sum
654:for every
466:for every
324:Properties
106:polynomial
45:references
2407:Resultant
2346:Trinomial
2326:Bivariate
2176:MathWorld
1940:…
1881:…
1498:…
1426:…
1267:∂
1259:∂
1216:∂
1208:∂
1175:∑
1122:…
1055:−
1029:−
993:−
957:−
946:−
727:…
662:λ
617:λ
611:…
595:λ
585:⇒
559:…
506:λ
474:λ
431:…
402:λ
382:λ
376:…
360:λ
70:July 2018
2392:Division
2341:Binomial
2336:Monomial
2111:(2005).
1975:See also
691:Given a
299:geometry
260:function
1552:is the
833:of the
829:is the
771:form a
755:over a
486:in any
343:, then
316:, as a
305:is the
258:, is a
102:quantic
58:improve
2229:degree
2121:
1554:degree
1548:where
1247:where
777:module
775:(or a
301:, the
110:degree
47:, but
2032:Notes
1678:then
1351:,...,
757:field
488:field
297:. In
283:field
276:basis
2218:and
2119:ISBN
2050:form
2048:and
761:ring
287:ring
268:form
256:form
247:.
96:, a
2227:By
1556:of
1297:of
910:in
902:in
864:).
494:of
285:or
250:An
92:In
2434::
2173:.
1371::
871:)
763:)
681:.
522:.
278:.
2208:e
2201:t
2194:v
2179:.
2127:.
1959:.
1956:)
1951:n
1947:x
1943:,
1937:,
1932:1
1928:x
1924:,
1921:1
1918:(
1914:P
1908:h
1900:=
1897:)
1892:n
1888:x
1884:,
1878:,
1873:1
1869:x
1865:(
1862:P
1849:0
1846:x
1829:.
1824:3
1819:0
1815:x
1811:7
1808:+
1803:2
1799:x
1793:1
1789:x
1783:0
1779:x
1775:+
1770:3
1765:3
1761:x
1757:=
1754:)
1749:3
1745:x
1741:,
1736:2
1732:x
1728:,
1723:1
1719:x
1715:,
1710:0
1706:x
1702:(
1699:P
1693:h
1663:,
1660:7
1657:+
1652:2
1648:x
1642:1
1638:x
1634:+
1629:3
1624:3
1620:x
1616:=
1613:)
1608:3
1604:x
1600:,
1595:2
1591:x
1587:,
1582:1
1578:x
1574:(
1571:P
1558:P
1550:d
1533:,
1529:)
1521:0
1517:x
1511:n
1507:x
1501:,
1495:,
1488:0
1484:x
1478:1
1474:x
1467:(
1463:P
1458:d
1453:0
1449:x
1445:=
1442:)
1437:n
1433:x
1429:,
1423:,
1418:1
1414:x
1410:,
1405:0
1401:x
1397:(
1393:P
1387:h
1369:P
1365:0
1362:x
1357:n
1353:x
1349:1
1346:x
1344:(
1342:P
1321:.
1316:i
1312:x
1300:P
1275:i
1271:x
1262:P
1232:,
1224:i
1220:x
1211:P
1200:i
1196:x
1190:n
1185:1
1182:=
1179:i
1171:=
1168:P
1165:d
1138:,
1133:n
1129:x
1125:,
1119:,
1114:1
1110:x
1098:d
1092:P
1070:.
1064:!
1061:)
1058:1
1052:n
1049:(
1046:!
1043:d
1038:!
1035:)
1032:1
1026:n
1023:+
1020:d
1017:(
1011:=
1005:)
1000:d
996:1
990:n
987:+
984:d
978:(
972:=
966:)
960:1
954:n
949:1
943:n
940:+
937:d
931:(
912:n
908:d
904:n
900:d
884:d
880:R
846:d
842:R
817:R
797:.
792:d
788:R
769:d
765:K
743:]
738:n
734:x
730:,
724:,
719:1
715:x
711:[
708:K
705:=
702:R
665:.
639:,
636:0
633:=
630:)
625:n
621:x
614:,
608:,
603:1
599:x
592:(
589:P
581:0
578:=
575:)
570:n
566:x
562:,
556:,
551:1
547:x
543:(
540:P
527:P
520:d
496:P
451:,
447:)
442:n
438:x
434:,
428:,
423:1
419:x
415:(
412:P
406:d
398:=
395:)
390:n
386:x
379:,
373:,
368:1
364:x
357:(
354:P
341:d
337:P
229:7
225:z
221:+
218:y
213:2
209:x
205:3
202:+
197:3
193:x
170:4
166:y
162:x
159:9
156:+
151:2
147:y
141:3
137:x
133:2
130:+
125:5
121:x
83:)
77:(
72:)
68:(
54:.
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.