2164:
1733:
2159:{\displaystyle 2-{\frac {\sqrt {{9\left({\frac {8{\sqrt {10}}i}{3^{3/2}}}+72\right)^{2/3}+36\left({\frac {8{\sqrt {10}}i}{3^{3/2}}}+72\right)^{1/3}+156} \over {\left({\frac {8{\sqrt {10}}i}{3^{3/2}}}+72\right)^{1/3}}}}{6}}-{{\sqrt {-\left({\frac {8{\sqrt {10}}i}{3^{3/2}}}+72\right)^{1/3}-{{52} \over {3\left({\frac {8{\sqrt {10}}i}{3^{3/2}}}+72\right)^{1/3}}}+8}} \over 2}.}
1190:
1408:
659:
955:
1672:
777:
1510:
1220:
433:
458:
1530:
1225:
960:
463:
939:
268:
1185:{\displaystyle {\begin{aligned}&x^{8}+4x^{7}+10x^{6}+16x^{5}+19x^{4}+16x^{3}+10x^{2}+4x-1\\={}&\left(x^{2}-2\right)\circ \left(x^{2}\right)\circ \left(x^{2}+x+1\right)\end{aligned}}}
1525:
1714:
193:
817:
672:
57:
2442:
869:
837:
839:. The restriction in the definition to polynomials of degree greater than one excludes the infinitely many decompositions possible with linear polynomials.
1403:{\displaystyle {\begin{aligned}&x^{6}-6x^{5}+15x^{4}-20x^{3}+15x^{2}-6x-1\\={}&\left(x^{3}-2\right)\circ \left(x^{2}-2x+1\right),\end{aligned}}}
1419:
279:
1519:, where there is an explicit formula for the roots, solving using the decomposition often gives a simpler form. For example, the decomposition
2250:
Jean-Charles Faugère, Ludovic Perret, "An efficient algorithm for decomposing multivariate polynomials and its applications to cryptography",
654:{\displaystyle {\begin{aligned}&x^{6}-6x^{5}+21x^{4}-44x^{3}+68x^{2}-64x+41\\={}&(x^{3}+9x^{2}+32x+41)\circ (x^{2}-2x).\end{aligned}}}
125:
The rest of this article discusses only univariate polynomials; algorithms also exist for multivariate polynomials of arbitrary degree.
2174:
The first algorithm for polynomial decomposition was published in 1985, though it had been discovered in 1976, and implemented in the
2470:
2252:
2179:
2426:
871:, and the degrees of the components are the same up to linear transformations, but possibly in different order; this is
115:
878:
204:
2494:
1667:{\displaystyle {\begin{aligned}&x^{4}-8x^{3}+18x^{2}-8x+2\\={}&(x^{2}+1)\circ (x^{2}-4x+1)\end{aligned}}}
2199:
A 2014 algorithm calculates a decomposition in polynomial time and without restrictions on the characteristic.
2186:
1211:
80:
2365:
2182:
1683:
2397:
1207:
111:
76:
31:
88:
143:
2489:
1724:
782:
446:
439:
2402:
772:{\displaystyle f=g_{1}\circ g_{2}\circ \cdots \circ g_{m}=h_{1}\circ h_{2}\circ \cdots \circ h_{n}}
2352:
2235:
1516:
1196:
2466:
2300:
949:
A polynomial decomposition may enable more efficient evaluation of a polynomial. For example,
36:
2407:
2334:
2284:
2261:
2228:
1720:
119:
2446:
2190:
92:
848:
1195:
can be calculated with 3 multiplications and 3 additions using the decomposition, while
822:
2427:"A polynomial time algorithm for computing all minimal decompositions of a polynomial"
2411:
2338:
2276:
Capi
Corrales-Rodrigáñez, "A note on Ritt's theorem on decomposition of polynomials",
2196:
A 1989 algorithm runs in polynomial time but with restrictions on the characteristic.
2185:. That algorithm takes exponential time in worst case, but works independently of the
2483:
2288:
669:
A polynomial may have distinct decompositions into indecomposable polynomials where
2385:
2381:
2217:
1505:{\displaystyle 1\pm 2^{1/6},1\pm {\frac {\sqrt {-1\pm {\sqrt {3}}i}}{2^{1/3}}}.}
842:
2325:
David R. Barton, Richard Zippel (1985). "Polynomial
Decomposition Algorithms".
2265:
84:
24:
428:{\displaystyle f(x)=(g\circ h)(x)=g(h(x))=g(x^{3})=(x^{3})^{2}-3(x^{3})+1,}
1203:
134:
2239:
2175:
1202:
A polynomial decomposition enables calculation of symbolic roots using
2232:
2463:
Computer
Algebra and Symbolic Computation: Mathematical Methods
1413:
the roots of this irreducible polynomial can be calculated as
2461:
1723:
gives equivalent results but in a form that is difficult to
122:, the product of the degrees of the composed polynomials.
1727:
and difficult to understand; one of the four roots is:
1736:
1686:
1528:
1422:
1223:
958:
881:
851:
825:
785:
675:
461:
282:
207:
146:
39:
118:). The degree of a composite polynomial is always a
98:Polynomials which are decomposable in this way are
2158:
1708:
1666:
1504:
1402:
1184:
933:
863:
831:
811:
771:
653:
427:
262:
187:
133:In the simplest case, one of the polynomials is a
51:
2222:Transactions of the American Mathematical Society
1199:would require 7 multiplications and 8 additions.
934:{\displaystyle x^{2}\circ x^{3}=x^{3}\circ x^{2}}
263:{\displaystyle g=x^{2}-3x+1{\text{ and }}h=x^{3}}
2388:(1989). "Polynomial Decomposition Algorithms".
8:
2401:
2299:The examples below were calculated using
2127:
2123:
2100:
2096:
2081:
2075:
2065:
2060:
2058:
2045:
2041:
2018:
2014:
1999:
1993:
1982:
1980:
1959:
1955:
1932:
1928:
1913:
1907:
1900:
1883:
1879:
1856:
1852:
1837:
1831:
1809:
1805:
1782:
1778:
1763:
1757:
1747:
1743:
1735:
1693:
1685:
1636:
1611:
1598:
1570:
1554:
1538:
1529:
1527:
1487:
1483:
1468:
1456:
1437:
1433:
1421:
1367:
1338:
1325:
1297:
1281:
1265:
1249:
1233:
1224:
1222:
1155:
1133:
1105:
1092:
1064:
1048:
1032:
1016:
1000:
984:
968:
959:
957:
925:
912:
899:
886:
880:
850:
824:
803:
790:
784:
763:
744:
731:
718:
699:
686:
674:
626:
592:
576:
563:
535:
519:
503:
487:
471:
462:
460:
407:
388:
378:
359:
281:
254:
239:
218:
206:
173:
157:
145:
38:
16:Factorization under function composition
2311:
2309:
2207:
1719:but straightforward application of the
1214:. For example, using the decomposition
873:Ritt's polynomial decomposition theorem
2434:ACM Communications in Computer Algebra
2213:
2211:
2220:, "Prime and Composite Polynomials",
1709:{\displaystyle 2\pm {\sqrt {3\pm i}}}
116:factored into products of polynomials
7:
2315:Where each ± is taken independently.
2278:Journal of Pure and Applied Algebra
79:greater than 1; it is an algebraic
14:
1210:. This technique is used in many
188:{\displaystyle f=x^{6}-3x^{3}+1}
2390:Journal of Symbolic Computation
2327:Journal of Symbolic Computation
2253:Journal of Symbolic Computation
812:{\displaystyle g_{i}\neq h_{i}}
1657:
1629:
1623:
1604:
641:
619:
613:
569:
413:
400:
385:
371:
365:
352:
343:
340:
334:
328:
319:
313:
310:
298:
292:
286:
1:
2412:10.1016/S0747-7171(89)80027-6
2339:10.1016/S0747-7171(85)80012-2
2283::3:293–296 (6 December 1990)
2289:10.1016/0022-4049(90)90086-W
2511:
104:indecomposable polynomials
102:; those which are not are
87:are known for decomposing
2266:10.1016/j.jsc.2008.02.005
2227::1:51–66 (January, 1922)
110:(not to be confused with
2425:Raoul Blankertz (2014).
2353:Functional Decomposition
1212:computer algebra systems
81:functional decomposition
52:{\displaystyle g\circ h}
21:polynomial decomposition
2183:computer algebra system
1208:irreducible polynomials
112:irreducible polynomials
2160:
1710:
1668:
1506:
1404:
1186:
935:
865:
833:
813:
773:
655:
429:
264:
189:
89:univariate polynomials
53:
32:functional composition
2161:
1711:
1669:
1507:
1405:
1187:
936:
866:
834:
814:
774:
656:
440:ring operator symbol
430:
265:
190:
100:composite polynomials
54:
1734:
1684:
1526:
1515:Even in the case of
1420:
1221:
956:
879:
849:
823:
783:
673:
459:
447:function composition
280:
205:
144:
37:
2260::1676-1689 (2009),
1517:quartic polynomials
864:{\displaystyle m=n}
2445:2015-09-24 at the
2189:of the underlying
2156:
1706:
1664:
1662:
1502:
1400:
1398:
1182:
1180:
931:
861:
829:
809:
769:
651:
649:
425:
260:
185:
114:, which cannot be
49:
19:In mathematics, a
2151:
2146:
2138:
2110:
2086:
2028:
2004:
1975:
1971:
1970:
1942:
1918:
1866:
1842:
1792:
1768:
1704:
1497:
1478:
1473:
832:{\displaystyle i}
242:
108:prime polynomials
2502:
2495:Computer algebra
2476:
2449:
2441:
2431:
2422:
2416:
2415:
2405:
2378:
2372:
2362:
2356:
2351:Richard Zippel,
2349:
2343:
2342:
2322:
2316:
2313:
2304:
2297:
2291:
2274:
2268:
2248:
2242:
2215:
2165:
2163:
2162:
2157:
2152:
2147:
2139:
2137:
2136:
2135:
2131:
2122:
2118:
2111:
2109:
2108:
2104:
2091:
2087:
2082:
2076:
2064:
2059:
2054:
2053:
2049:
2040:
2036:
2029:
2027:
2026:
2022:
2009:
2005:
2000:
1994:
1983:
1981:
1976:
1969:
1968:
1967:
1963:
1954:
1950:
1943:
1941:
1940:
1936:
1923:
1919:
1914:
1908:
1899:
1892:
1891:
1887:
1878:
1874:
1867:
1865:
1864:
1860:
1847:
1843:
1838:
1832:
1818:
1817:
1813:
1804:
1800:
1793:
1791:
1790:
1786:
1773:
1769:
1764:
1758:
1746:
1745:
1744:
1715:
1713:
1712:
1707:
1705:
1694:
1677:gives the roots
1673:
1671:
1670:
1665:
1663:
1641:
1640:
1616:
1615:
1599:
1575:
1574:
1559:
1558:
1543:
1542:
1532:
1511:
1509:
1508:
1503:
1498:
1496:
1495:
1491:
1474:
1469:
1458:
1457:
1446:
1445:
1441:
1409:
1407:
1406:
1401:
1399:
1392:
1388:
1372:
1371:
1354:
1350:
1343:
1342:
1326:
1302:
1301:
1286:
1285:
1270:
1269:
1254:
1253:
1238:
1237:
1227:
1206:, even for some
1191:
1189:
1188:
1183:
1181:
1177:
1173:
1160:
1159:
1142:
1138:
1137:
1121:
1117:
1110:
1109:
1093:
1069:
1068:
1053:
1052:
1037:
1036:
1021:
1020:
1005:
1004:
989:
988:
973:
972:
962:
940:
938:
937:
932:
930:
929:
917:
916:
904:
903:
891:
890:
870:
868:
867:
862:
838:
836:
835:
830:
818:
816:
815:
810:
808:
807:
795:
794:
778:
776:
775:
770:
768:
767:
749:
748:
736:
735:
723:
722:
704:
703:
691:
690:
660:
658:
657:
652:
650:
631:
630:
597:
596:
581:
580:
564:
540:
539:
524:
523:
508:
507:
492:
491:
476:
475:
465:
452:Less trivially,
434:
432:
431:
426:
412:
411:
393:
392:
383:
382:
364:
363:
269:
267:
266:
261:
259:
258:
243:
240:
223:
222:
198:decomposes into
194:
192:
191:
186:
178:
177:
162:
161:
120:composite number
58:
56:
55:
50:
2510:
2509:
2505:
2504:
2503:
2501:
2500:
2499:
2480:
2479:
2473:
2460:
2457:
2452:
2447:Wayback Machine
2429:
2424:
2423:
2419:
2403:10.1.1.416.6491
2380:
2379:
2375:
2363:
2359:
2350:
2346:
2324:
2323:
2319:
2314:
2307:
2298:
2294:
2275:
2271:
2249:
2245:
2233:10.2307/1988911
2216:
2209:
2205:
2172:
2092:
2077:
2074:
2070:
2069:
2010:
1995:
1992:
1988:
1987:
1924:
1909:
1906:
1902:
1901:
1848:
1833:
1830:
1826:
1825:
1774:
1759:
1756:
1752:
1751:
1732:
1731:
1721:quartic formula
1682:
1681:
1661:
1660:
1632:
1607:
1600:
1592:
1591:
1566:
1550:
1534:
1524:
1523:
1479:
1429:
1418:
1417:
1397:
1396:
1363:
1362:
1358:
1334:
1333:
1329:
1327:
1319:
1318:
1293:
1277:
1261:
1245:
1229:
1219:
1218:
1197:Horner's method
1179:
1178:
1151:
1150:
1146:
1129:
1125:
1101:
1100:
1096:
1094:
1086:
1085:
1060:
1044:
1028:
1012:
996:
980:
964:
954:
953:
947:
921:
908:
895:
882:
877:
876:
875:. For example,
847:
846:
821:
820:
799:
786:
781:
780:
759:
740:
727:
714:
695:
682:
671:
670:
667:
648:
647:
622:
588:
572:
565:
557:
556:
531:
515:
499:
483:
467:
457:
456:
403:
384:
374:
355:
278:
277:
250:
241: and
214:
203:
202:
169:
153:
142:
141:
137:. For example,
131:
93:polynomial time
59:of polynomials
35:
34:
17:
12:
11:
5:
2508:
2506:
2498:
2497:
2492:
2482:
2481:
2478:
2477:
2471:
2456:
2453:
2451:
2450:
2417:
2396:(5): 445–456.
2373:
2357:
2344:
2333:(2): 159–168.
2317:
2305:
2292:
2269:
2243:
2206:
2204:
2201:
2187:characteristic
2171:
2168:
2167:
2166:
2155:
2150:
2145:
2142:
2134:
2130:
2126:
2121:
2117:
2114:
2107:
2103:
2099:
2095:
2090:
2085:
2080:
2073:
2068:
2063:
2057:
2052:
2048:
2044:
2039:
2035:
2032:
2025:
2021:
2017:
2013:
2008:
2003:
1998:
1991:
1986:
1979:
1974:
1966:
1962:
1958:
1953:
1949:
1946:
1939:
1935:
1931:
1927:
1922:
1917:
1912:
1905:
1898:
1895:
1890:
1886:
1882:
1877:
1873:
1870:
1863:
1859:
1855:
1851:
1846:
1841:
1836:
1829:
1824:
1821:
1816:
1812:
1808:
1803:
1799:
1796:
1789:
1785:
1781:
1777:
1772:
1767:
1762:
1755:
1750:
1742:
1739:
1717:
1716:
1703:
1700:
1697:
1692:
1689:
1675:
1674:
1659:
1656:
1653:
1650:
1647:
1644:
1639:
1635:
1631:
1628:
1625:
1622:
1619:
1614:
1610:
1606:
1603:
1601:
1597:
1594:
1593:
1590:
1587:
1584:
1581:
1578:
1573:
1569:
1565:
1562:
1557:
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1549:
1546:
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1537:
1533:
1531:
1513:
1512:
1501:
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1410:
1395:
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1387:
1384:
1381:
1378:
1375:
1370:
1366:
1361:
1357:
1353:
1349:
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1328:
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1320:
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1300:
1296:
1292:
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1276:
1273:
1268:
1264:
1260:
1257:
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1241:
1236:
1232:
1228:
1226:
1193:
1192:
1176:
1172:
1169:
1166:
1163:
1158:
1154:
1149:
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1141:
1136:
1132:
1128:
1124:
1120:
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1113:
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1063:
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1051:
1047:
1043:
1040:
1035:
1031:
1027:
1024:
1019:
1015:
1011:
1008:
1003:
999:
995:
992:
987:
983:
979:
976:
971:
967:
963:
961:
946:
943:
928:
924:
920:
915:
911:
907:
902:
898:
894:
889:
885:
860:
857:
854:
828:
806:
802:
798:
793:
789:
766:
762:
758:
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752:
747:
743:
739:
734:
730:
726:
721:
717:
713:
710:
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702:
698:
694:
689:
685:
681:
678:
666:
663:
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661:
646:
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634:
629:
625:
621:
618:
615:
612:
609:
606:
603:
600:
595:
591:
587:
584:
579:
575:
571:
568:
566:
562:
559:
558:
555:
552:
549:
546:
543:
538:
534:
530:
527:
522:
518:
514:
511:
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502:
498:
495:
490:
486:
482:
479:
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464:
436:
435:
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421:
418:
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410:
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127:
48:
45:
42:
15:
13:
10:
9:
6:
4:
3:
2:
2507:
2496:
2493:
2491:
2488:
2487:
2485:
2474:
2472:1-56881-159-4
2468:
2464:
2459:
2458:
2454:
2448:
2444:
2439:
2435:
2428:
2421:
2418:
2413:
2409:
2404:
2399:
2395:
2391:
2387:
2386:Landau, Susan
2383:
2382:Kozen, Dexter
2377:
2374:
2370:
2368:
2361:
2358:
2354:
2348:
2345:
2340:
2336:
2332:
2328:
2321:
2318:
2312:
2310:
2306:
2302:
2296:
2293:
2290:
2286:
2282:
2279:
2273:
2270:
2267:
2263:
2259:
2255:
2254:
2247:
2244:
2241:
2237:
2234:
2230:
2226:
2223:
2219:
2214:
2212:
2208:
2202:
2200:
2197:
2194:
2192:
2188:
2184:
2181:
2177:
2169:
2153:
2148:
2143:
2140:
2132:
2128:
2124:
2119:
2115:
2112:
2105:
2101:
2097:
2093:
2088:
2083:
2078:
2071:
2066:
2061:
2055:
2050:
2046:
2042:
2037:
2033:
2030:
2023:
2019:
2015:
2011:
2006:
2001:
1996:
1989:
1984:
1977:
1972:
1964:
1960:
1956:
1951:
1947:
1944:
1937:
1933:
1929:
1925:
1920:
1915:
1910:
1903:
1896:
1893:
1888:
1884:
1880:
1875:
1871:
1868:
1861:
1857:
1853:
1849:
1844:
1839:
1834:
1827:
1822:
1819:
1814:
1810:
1806:
1801:
1797:
1794:
1787:
1783:
1779:
1775:
1770:
1765:
1760:
1753:
1748:
1740:
1737:
1730:
1729:
1728:
1726:
1722:
1701:
1698:
1695:
1690:
1687:
1680:
1679:
1678:
1654:
1651:
1648:
1645:
1642:
1637:
1633:
1626:
1620:
1617:
1612:
1608:
1602:
1595:
1588:
1585:
1582:
1579:
1576:
1571:
1567:
1563:
1560:
1555:
1551:
1547:
1544:
1539:
1535:
1522:
1521:
1520:
1518:
1499:
1492:
1488:
1484:
1480:
1475:
1470:
1465:
1462:
1459:
1453:
1450:
1447:
1442:
1438:
1434:
1430:
1426:
1423:
1416:
1415:
1414:
1393:
1389:
1385:
1382:
1379:
1376:
1373:
1368:
1364:
1359:
1355:
1351:
1347:
1344:
1339:
1335:
1330:
1322:
1315:
1312:
1309:
1306:
1303:
1298:
1294:
1290:
1287:
1282:
1278:
1274:
1271:
1266:
1262:
1258:
1255:
1250:
1246:
1242:
1239:
1234:
1230:
1217:
1216:
1215:
1213:
1209:
1205:
1200:
1198:
1174:
1170:
1167:
1164:
1161:
1156:
1152:
1147:
1143:
1139:
1134:
1130:
1126:
1122:
1118:
1114:
1111:
1106:
1102:
1097:
1089:
1082:
1079:
1076:
1073:
1070:
1065:
1061:
1057:
1054:
1049:
1045:
1041:
1038:
1033:
1029:
1025:
1022:
1017:
1013:
1009:
1006:
1001:
997:
993:
990:
985:
981:
977:
974:
969:
965:
952:
951:
950:
944:
942:
926:
922:
918:
913:
909:
905:
900:
896:
892:
887:
883:
874:
858:
855:
852:
844:
840:
826:
804:
800:
796:
791:
787:
764:
760:
756:
753:
750:
745:
741:
737:
732:
728:
724:
719:
715:
711:
708:
705:
700:
696:
692:
687:
683:
679:
676:
664:
644:
638:
635:
632:
627:
623:
616:
610:
607:
604:
601:
598:
593:
589:
585:
582:
577:
573:
567:
560:
553:
550:
547:
544:
541:
536:
532:
528:
525:
520:
516:
512:
509:
504:
500:
496:
493:
488:
484:
480:
477:
472:
468:
455:
454:
453:
450:
448:
444:
443:
422:
419:
416:
408:
404:
397:
394:
389:
379:
375:
368:
360:
356:
349:
346:
337:
331:
325:
322:
316:
307:
304:
301:
295:
289:
283:
276:
275:
274:
255:
251:
247:
244:
236:
233:
230:
227:
224:
219:
215:
211:
208:
201:
200:
199:
182:
179:
174:
170:
166:
163:
158:
154:
150:
147:
140:
139:
138:
136:
128:
126:
123:
121:
117:
113:
109:
106:or sometimes
105:
101:
96:
94:
90:
86:
82:
78:
74:
70:
66:
62:
46:
43:
40:
33:
29:
26:
22:
2462:
2437:
2433:
2420:
2393:
2389:
2376:
2366:
2360:
2347:
2330:
2326:
2320:
2295:
2280:
2277:
2272:
2257:
2251:
2246:
2224:
2221:
2198:
2195:
2173:
1718:
1676:
1514:
1412:
1201:
1194:
948:
945:Applications
872:
845:proved that
841:
668:
451:
441:
437:
272:
197:
132:
124:
107:
103:
99:
97:
72:
68:
64:
60:
27:
23:expresses a
20:
18:
2490:Polynomials
843:Joseph Ritt
2484:Categories
2455:References
2367:polydecomp
2170:Algorithms
665:Uniqueness
445:to denote
438:using the
85:Algorithms
25:polynomial
2440:(187): 1.
2398:CiteSeerX
2218:J.F. Ritt
2056:−
1985:−
1978:−
1741:−
1699:±
1691:±
1643:−
1627:∘
1577:−
1545:−
1466:±
1460:−
1454:±
1427:±
1374:−
1356:∘
1345:−
1313:−
1304:−
1272:−
1240:−
1144:∘
1123:∘
1112:−
1080:−
919:∘
893:∘
819:for some
797:≠
757:∘
754:⋯
751:∘
738:∘
712:∘
709:⋯
706:∘
693:∘
633:−
617:∘
542:−
510:−
478:−
395:−
305:∘
225:−
164:−
44:∘
2443:Archived
2369:function
2364:See the
1725:simplify
1204:radicals
135:monomial
129:Examples
67:, where
2355:, 1996.
2240:1988911
2176:Macsyma
30:as the
2469:
2400:
2301:Maxima
2238:
2180:Maxima
779:where
273:since
77:degree
2430:(PDF)
2236:JSTOR
2203:Notes
2191:field
75:have
2467:ISBN
71:and
63:and
2408:doi
2335:doi
2285:doi
2262:doi
2229:doi
1897:156
91:in
2486::
2465:.
2438:48
2436:.
2432:.
2406:.
2392:.
2384:;
2329:.
2308:^
2281:68
2258:44
2256:,
2225:23
2210:^
2193:.
2116:72
2084:10
2062:52
2034:72
2002:10
1948:72
1916:10
1872:72
1840:10
1823:36
1798:72
1766:10
1564:18
1291:15
1275:20
1259:15
1058:10
1042:16
1026:19
1010:16
994:10
941:.
611:41
602:32
554:41
545:64
529:68
513:44
497:21
449:.
95:.
83:.
2475:.
2414:.
2410::
2394:7
2371:.
2341:.
2337::
2331:1
2303:.
2287::
2264::
2231::
2178:/
2154:.
2149:2
2144:8
2141:+
2133:3
2129:/
2125:1
2120:)
2113:+
2106:2
2102:/
2098:3
2094:3
2089:i
2079:8
2072:(
2067:3
2051:3
2047:/
2043:1
2038:)
2031:+
2024:2
2020:/
2016:3
2012:3
2007:i
1997:8
1990:(
1973:6
1965:3
1961:/
1957:1
1952:)
1945:+
1938:2
1934:/
1930:3
1926:3
1921:i
1911:8
1904:(
1894:+
1889:3
1885:/
1881:1
1876:)
1869:+
1862:2
1858:/
1854:3
1850:3
1845:i
1835:8
1828:(
1820:+
1815:3
1811:/
1807:2
1802:)
1795:+
1788:2
1784:/
1780:3
1776:3
1771:i
1761:8
1754:(
1749:9
1738:2
1702:i
1696:3
1688:2
1658:)
1655:1
1652:+
1649:x
1646:4
1638:2
1634:x
1630:(
1624:)
1621:1
1618:+
1613:2
1609:x
1605:(
1596:=
1589:2
1586:+
1583:x
1580:8
1572:2
1568:x
1561:+
1556:3
1552:x
1548:8
1540:4
1536:x
1500:.
1493:3
1489:/
1485:1
1481:2
1476:i
1471:3
1463:1
1451:1
1448:,
1443:6
1439:/
1435:1
1431:2
1424:1
1394:,
1390:)
1386:1
1383:+
1380:x
1377:2
1369:2
1365:x
1360:(
1352:)
1348:2
1340:3
1336:x
1331:(
1323:=
1316:1
1310:x
1307:6
1299:2
1295:x
1288:+
1283:3
1279:x
1267:4
1263:x
1256:+
1251:5
1247:x
1243:6
1235:6
1231:x
1175:)
1171:1
1168:+
1165:x
1162:+
1157:2
1153:x
1148:(
1140:)
1135:2
1131:x
1127:(
1119:)
1115:2
1107:2
1103:x
1098:(
1090:=
1083:1
1077:x
1074:4
1071:+
1066:2
1062:x
1055:+
1050:3
1046:x
1039:+
1034:4
1030:x
1023:+
1018:5
1014:x
1007:+
1002:6
998:x
991:+
986:7
982:x
978:4
975:+
970:8
966:x
927:2
923:x
914:3
910:x
906:=
901:3
897:x
888:2
884:x
859:n
856:=
853:m
827:i
805:i
801:h
792:i
788:g
765:n
761:h
746:2
742:h
733:1
729:h
725:=
720:m
716:g
701:2
697:g
688:1
684:g
680:=
677:f
645:.
642:)
639:x
636:2
628:2
624:x
620:(
614:)
608:+
605:x
599:+
594:2
590:x
586:9
583:+
578:3
574:x
570:(
561:=
551:+
548:x
537:2
533:x
526:+
521:3
517:x
505:4
501:x
494:+
489:5
485:x
481:6
473:6
469:x
442:∘
423:,
420:1
417:+
414:)
409:3
405:x
401:(
398:3
390:2
386:)
380:3
376:x
372:(
369:=
366:)
361:3
357:x
353:(
350:g
347:=
344:)
341:)
338:x
335:(
332:h
329:(
326:g
323:=
320:)
317:x
314:(
311:)
308:h
302:g
299:(
296:=
293:)
290:x
287:(
284:f
256:3
252:x
248:=
245:h
237:1
234:+
231:x
228:3
220:2
216:x
212:=
209:g
183:1
180:+
175:3
171:x
167:3
159:6
155:x
151:=
148:f
73:h
69:g
65:h
61:g
47:h
41:g
28:f
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