1237:
922:
658:
Both uses of this notion can be found, and in many cases the distinction is simply ignored, see for instance examples for the first and second meaning. In informal discussions the distinction is seldom important, and tendency is towards the broader second meaning. When studying the structure of
2172:
The degree of a monomial is defined as the sum of all the exponents of the variables, including the implicit exponents of 1 for the variables which appear without exponent; e.g., in the example of the previous section, the degree is
1232:{\displaystyle \left(\!\!{\binom {n}{d}}\!\!\right)={\binom {n+d-1}{d}}={\binom {d+(n-1)}{n-1}}={\frac {(d+1)\times (d+2)\times \cdots \times (d+n-1)}{1\times 2\times \cdots \times (n-1)}}={\frac {1}{(n-1)!}}(d+1)^{\overline {n-1}}.}
1468:
2162:
2241:
The degree of a monomial is sometimes called order, mainly in the context of series. It is also called total degree when it is needed to distinguish it from the degree in one of the variables.
2061:
675:
is used with the first meaning, but it does not make the absence of constants clear either), while the notion term of a polynomial unambiguously coincides with the second meaning of monomial.
856:
1692:
1931:
671:
of that basis. An argument in favor of the first meaning is also that no obvious other notion is available to designate these values (the term power product is in use, in particular when
1854:
1990:
1597:
571:
121:
1338:
381:
2307:
2097:
2236:
513:
606:
334:
2203:
413:
259:
172:
1794:
is often useful for having a compact notation, specially when there are more than two or three variables. If the variables being used form an indexed family like
1738:
1367:
1294:
914:
1778:
1758:
1712:
1539:
1519:
1499:
1268:
884:
800:
780:
756:
736:
480:
453:
433:
299:
279:
232:
212:
192:
141:
1242:
The latter forms are particularly useful when one fixes the number of variables and lets the degree vary. From these expressions one sees that for fixed
1376:
2480:
2660:
2422:
635:
Since the word "monomial", as well as the word "polynomial", comes from the late Latin word "binomium" (binomial), by changing the
659:
polynomials however, one often definitely needs a notion with the first meaning. This is for instance the case when considering a
2711:
2108:
2244:
Monomial degree is fundamental to the theory of univariate and multivariate polynomials. Explicitly, it is used to define the
2345:
462:
of the monomial. A monomial in the first sense is a special case of a monomial in the second sense, where the coefficient is
2448:
2655:
2639:
2001:
2675:
2443:
808:
1617:
1862:
2520:
2314:
2473:
655:
With either definition, the set of monomials is a subset of all polynomials that is closed under multiplication.
2102:
With this notation, the product of two monomials is simply expressed by using the addition of exponent vectors:
1797:
2665:
1942:
1551:
2604:
2340:
2634:
2629:
2624:
2502:
2350:
2249:
2245:
699:
55:
2414:
1481:
is a compact way to express the number of monomials of a given degree: the number of monomials of degree
2614:
2594:
863:
518:
65:
802:
variables (a variable can be chosen more than once, but order does not matter), which is given by the
2680:
2599:
2466:
2355:
1790:
1299:
859:
803:
339:
2493:
2279:
1542:
640:
59:
2072:
2537:
2532:
2408:
2309:
for some set of α have special properties of homogeneity. This can be phrased in the language of
2273:
695:
617:
2670:
639:"bi-" (two in Latin), a monomial should theoretically be called a "mononomial". "Monomial" is a
62:
exponents, or, in other words, a product of variables, possibly with repetitions. For example,
2515:
2438:
2418:
2375:
2253:
1471:
887:
2208:
485:
2716:
2690:
2568:
2561:
2556:
2360:
2257:
759:
576:
304:
2176:
386:
237:
150:
2578:
2573:
2525:
2510:
2322:
2318:
2310:
664:
629:
1717:
1346:
1273:
893:
458:
A monomial is a monomial in the first sense multiplied by a nonzero constant, called the
2238:
is 1+1+2=4. The degree of a nonzero constant is 0. For example, the degree of −7 is 0.
2549:
2544:
2458:
2327:
1763:
1743:
1697:
1524:
1504:
1484:
1478:
1253:
869:
785:
765:
741:
721:
689:
668:
660:
625:
621:
609:
465:
438:
418:
284:
264:
217:
197:
177:
126:
2705:
2261:
144:
2685:
2365:
703:
1694:. This follows from the one-to-one correspondence between the monomials of degree
694:
The most obvious fact about monomials (first meaning) is that any polynomial is a
459:
31:
2489:
39:
2619:
2370:
644:
17:
1463:{\textstyle {\frac {1}{2}}(d+1)^{\overline {2}}={\frac {1}{2}}(d+1)(d+2)}
336:
then each can be given an exponent, so that any monomial is of the form
43:
679:
The remainder of this article assumes the first meaning of "monomial".
636:
624:, the exponents of a monomial may be negative, and in the context of
1780:
variables, which consists in substituting by 1 the extra variable.
2462:
301:
a positive integer. If several variables are considered, say,
1470:; these numbers form the sequence 1, 3, 6, 10, 15, ... of
2157:{\displaystyle x^{\alpha }x^{\beta }=x^{\alpha +\beta }.}
1343:
For example, the number of monomials in three variables (
573:
are monomials (in the second example, the variables are
1620:
1379:
1302:
811:
2282:
2211:
2179:
2111:
2075:
2004:
1945:
1865:
1800:
1766:
1746:
1720:
1700:
1554:
1527:
1507:
1487:
1349:
1276:
1256:
925:
896:
872:
858:. This expression can also be given in the form of a
788:
768:
744:
724:
579:
521:
488:
468:
441:
421:
415:
non-negative integers (taking note that any exponent
389:
342:
307:
287:
267:
240:
220:
200:
180:
153:
129:
68:
2395:
American
Heritage Dictionary of the English Language
2260:. Implicitly, it is used in grouping the terms of a
214:
is considered, this means that a monomial is either
46:. Two definitions of a monomial may be encountered:
2648:
2587:
2500:
2301:
2230:
2197:
2156:
2091:
2056:{\displaystyle x_{1}^{a}x_{2}^{b}x_{3}^{c}\cdots }
2055:
1984:
1925:
1848:
1772:
1752:
1732:
1706:
1686:
1591:
1533:
1513:
1493:
1462:
1361:
1332:
1288:
1262:
1231:
908:
878:
850:
794:
774:
750:
730:
710:- a fact of constant implicit use in mathematics.
600:
565:
507:
474:
447:
427:
407:
375:
328:
293:
273:
253:
226:
206:
186:
166:
135:
115:
1048:
1007:
995:
968:
956:
955:
949:
936:
932:
931:
851:{\textstyle \left(\!\!{\binom {n}{d}}\!\!\right)}
842:
841:
818:
817:
1687:{\textstyle {\binom {n+d}{n}}={\binom {n+d}{d}}}
2511:Zero polynomial (degree undefined or −1 or −∞)
2407:Cox, David; John Little; Donal O'Shea (1998).
1926:{\displaystyle x=(x_{1},x_{2},x_{3},\ldots ),}
1740:variables and the monomials of degree at most
2474:
1678:
1657:
1645:
1624:
835:
822:
8:
2276:the varieties defined by monomial equations
2321:(equivalently by a multiplicative group of
2481:
2467:
2459:
2325:). This area is studied under the name of
1849:{\displaystyle x_{1},x_{2},x_{3},\ldots ,}
1602:The number of monomials of degree at most
2287:
2281:
2222:
2210:
2178:
2139:
2126:
2116:
2110:
2080:
2074:
2044:
2039:
2029:
2024:
2014:
2009:
2003:
1944:
1905:
1892:
1879:
1864:
1831:
1818:
1805:
1799:
1765:
1745:
1719:
1699:
1677:
1656:
1654:
1644:
1623:
1621:
1619:
1577:
1555:
1553:
1526:
1506:
1486:
1420:
1406:
1380:
1378:
1348:
1303:
1301:
1275:
1255:
1207:
1164:
1057:
1047:
1006:
1004:
994:
967:
965:
948:
935:
933:
924:
895:
871:
834:
821:
819:
810:
787:
767:
743:
723:
578:
557:
544:
520:
499:
487:
467:
440:
420:
388:
367:
357:
347:
341:
306:
286:
266:
245:
239:
219:
199:
179:
158:
152:
128:
86:
73:
67:
1985:{\displaystyle \alpha =(a,b,c,\ldots ).}
435:makes the corresponding factor equal to
2387:
1592:{\displaystyle {\frac {1}{(1-t)^{n}}}.}
1521:variables is the coefficient of degree
482:. For example, in this interpretation
7:
1246:, the number of monomials of degree
2262:Taylor series in several variables
2256:used in formulating and computing
1661:
1628:
1011:
972:
940:
826:
718:The number of monomials of degree
566:{\displaystyle (3-4i)x^{4}yz^{13}}
143:is a monomial, being equal to the
116:{\displaystyle x^{2}yz^{3}=xxyzzz}
25:
2313:, in terms of the existence of a
651:Comparison of the two definitions
1333:{\textstyle {\frac {1}{(n-1)!}}}
706:of all polynomials, called the
376:{\displaystyle x^{a}y^{b}z^{c}}
1976:
1952:
1917:
1872:
1574:
1561:
1457:
1445:
1442:
1430:
1403:
1390:
1321:
1309:
1250:is a polynomial expression in
1204:
1191:
1182:
1170:
1155:
1143:
1120:
1102:
1090:
1078:
1072:
1060:
1031:
1019:
537:
522:
1:
2671:Horner's method of evaluation
2302:{\displaystyle x^{\alpha }=0}
27:Polynomial with only one term
2413:. Springer Verlag. pp.
2092:{\displaystyle x^{\alpha }.}
2066:can be compactly written as
1411:
1220:
194:. If only a single variable
123:is a monomial. The constant
54:, is a product of powers of
2676:Polynomial identity testing
2444:Encyclopedia of Mathematics
758:variables is the number of
2733:
782:elements chosen among the
687:
1296:with leading coefficient
608:and the coefficient is a
2410:Using Algebraic Geometry
2252:, as well as for graded
698:of them, so they form a
50:A monomial, also called
38:is, roughly speaking, a
2712:Homogeneous polynomials
2661:Greatest common divisor
2341:Monomial representation
2231:{\displaystyle xyz^{2}}
628:, the exponents may be
508:{\displaystyle -7x^{5}}
2533:Quadratic function (2)
2351:Homogeneous polynomial
2303:
2250:homogeneous polynomial
2246:degree of a polynomial
2232:
2199:
2158:
2093:
2057:
1986:
1927:
1850:
1774:
1754:
1734:
1708:
1688:
1593:
1535:
1515:
1495:
1464:
1363:
1334:
1290:
1264:
1233:
910:
888:rising factorial power
880:
852:
796:
776:
752:
732:
602:
601:{\displaystyle x,y,z,}
567:
509:
476:
449:
429:
409:
377:
330:
329:{\displaystyle x,y,z,}
295:
275:
255:
228:
208:
188:
168:
137:
117:
2516:Constant function (0)
2304:
2233:
2200:
2198:{\displaystyle a+b+c}
2159:
2094:
2058:
1987:
1928:
1851:
1775:
1755:
1735:
1709:
1689:
1594:
1536:
1516:
1496:
1465:
1364:
1335:
1291:
1265:
1234:
911:
881:
864:polynomial expression
853:
797:
777:
753:
733:
603:
568:
510:
477:
450:
430:
410:
408:{\displaystyle a,b,c}
378:
331:
296:
276:
256:
254:{\displaystyle x^{n}}
229:
209:
189:
169:
167:{\displaystyle x^{0}}
138:
118:
2649:Tools and algorithms
2569:Quintic function (5)
2557:Quartic function (4)
2494:polynomial functions
2356:Homogeneous function
2280:
2209:
2177:
2109:
2073:
2002:
1943:
1863:
1798:
1791:multi-index notation
1784:Multi-index notation
1764:
1744:
1718:
1698:
1618:
1552:
1525:
1505:
1485:
1377:
1347:
1300:
1274:
1254:
923:
894:
870:
860:binomial coefficient
809:
804:multiset coefficient
786:
766:
742:
722:
577:
519:
486:
466:
439:
419:
387:
340:
305:
285:
265:
238:
218:
198:
178:
151:
127:
66:
2579:Septic equation (7)
2574:Sextic equation (6)
2521:Linear function (1)
2049:
2034:
2019:
1995:Then the monomial
1733:{\displaystyle n+1}
1543:formal power series
1362:{\displaystyle n=3}
1289:{\displaystyle n-1}
909:{\displaystyle d+1}
618:Laurent polynomials
60:nonnegative integer
42:which has only one
2545:Cubic function (3)
2538:Quadratic equation
2299:
2274:algebraic geometry
2254:monomial orderings
2248:and the notion of
2228:
2195:
2154:
2089:
2053:
2035:
2020:
2005:
1982:
1923:
1846:
1770:
1750:
1730:
1704:
1684:
1589:
1531:
1511:
1491:
1472:triangular numbers
1460:
1359:
1330:
1286:
1260:
1229:
906:
876:
848:
792:
772:
748:
728:
696:linear combination
616:In the context of
598:
563:
505:
472:
445:
425:
405:
373:
326:
291:
271:
251:
224:
204:
184:
164:
133:
113:
2699:
2698:
2640:Quasi-homogeneous
2376:Sparse polynomial
2323:diagonal matrices
1773:{\displaystyle n}
1753:{\displaystyle d}
1707:{\displaystyle d}
1676:
1643:
1584:
1534:{\displaystyle d}
1514:{\displaystyle n}
1494:{\displaystyle d}
1428:
1414:
1388:
1328:
1263:{\displaystyle d}
1223:
1189:
1159:
1046:
993:
947:
879:{\displaystyle d}
833:
795:{\displaystyle n}
775:{\displaystyle d}
760:multicombinations
751:{\displaystyle n}
731:{\displaystyle d}
669:monomial ordering
647:of "mononomial".
475:{\displaystyle 1}
448:{\displaystyle 1}
428:{\displaystyle 0}
294:{\displaystyle n}
274:{\displaystyle x}
227:{\displaystyle 1}
207:{\displaystyle x}
187:{\displaystyle x}
174:for any variable
136:{\displaystyle 1}
16:(Redirected from
2724:
2562:Quartic equation
2483:
2476:
2469:
2460:
2453:
2452:
2435:
2429:
2428:
2404:
2398:
2392:
2361:Multilinear form
2328:torus embeddings
2311:algebraic groups
2308:
2306:
2305:
2300:
2292:
2291:
2237:
2235:
2234:
2229:
2227:
2226:
2205:. The degree of
2204:
2202:
2201:
2196:
2163:
2161:
2160:
2155:
2150:
2149:
2131:
2130:
2121:
2120:
2098:
2096:
2095:
2090:
2085:
2084:
2062:
2060:
2059:
2054:
2048:
2043:
2033:
2028:
2018:
2013:
1991:
1989:
1988:
1983:
1932:
1930:
1929:
1924:
1910:
1909:
1897:
1896:
1884:
1883:
1855:
1853:
1852:
1847:
1836:
1835:
1823:
1822:
1810:
1809:
1779:
1777:
1776:
1771:
1759:
1757:
1756:
1751:
1739:
1737:
1736:
1731:
1713:
1711:
1710:
1705:
1693:
1691:
1690:
1685:
1683:
1682:
1681:
1672:
1660:
1650:
1649:
1648:
1639:
1627:
1613:
1607:
1598:
1596:
1595:
1590:
1585:
1583:
1582:
1581:
1556:
1540:
1538:
1537:
1532:
1520:
1518:
1517:
1512:
1500:
1498:
1497:
1492:
1469:
1467:
1466:
1461:
1429:
1421:
1416:
1415:
1407:
1389:
1381:
1368:
1366:
1365:
1360:
1339:
1337:
1336:
1331:
1329:
1327:
1304:
1295:
1293:
1292:
1287:
1269:
1267:
1266:
1261:
1238:
1236:
1235:
1230:
1225:
1224:
1219:
1208:
1190:
1188:
1165:
1160:
1158:
1123:
1058:
1053:
1052:
1051:
1045:
1034:
1010:
1000:
999:
998:
989:
971:
961:
957:
954:
953:
952:
939:
915:
913:
912:
907:
885:
883:
882:
877:
857:
855:
854:
849:
847:
843:
840:
839:
838:
825:
801:
799:
798:
793:
781:
779:
778:
773:
757:
755:
754:
749:
737:
735:
734:
729:
630:rational numbers
607:
605:
604:
599:
572:
570:
569:
564:
562:
561:
549:
548:
514:
512:
511:
506:
504:
503:
481:
479:
478:
473:
454:
452:
451:
446:
434:
432:
431:
426:
414:
412:
411:
406:
382:
380:
379:
374:
372:
371:
362:
361:
352:
351:
335:
333:
332:
327:
300:
298:
297:
292:
280:
278:
277:
272:
260:
258:
257:
252:
250:
249:
233:
231:
230:
225:
213:
211:
210:
205:
193:
191:
190:
185:
173:
171:
170:
165:
163:
162:
142:
140:
139:
134:
122:
120:
119:
114:
91:
90:
78:
77:
21:
2732:
2731:
2727:
2726:
2725:
2723:
2722:
2721:
2702:
2701:
2700:
2695:
2644:
2583:
2526:Linear equation
2496:
2487:
2457:
2456:
2437:
2436:
2432:
2425:
2406:
2405:
2401:
2393:
2389:
2384:
2346:Monomial matrix
2337:
2319:algebraic torus
2283:
2278:
2277:
2270:
2218:
2207:
2206:
2175:
2174:
2170:
2135:
2122:
2112:
2107:
2106:
2076:
2071:
2070:
2000:
1999:
1941:
1940:
1901:
1888:
1875:
1861:
1860:
1827:
1814:
1801:
1796:
1795:
1786:
1762:
1761:
1742:
1741:
1716:
1715:
1696:
1695:
1662:
1655:
1629:
1622:
1616:
1615:
1609:
1603:
1573:
1560:
1550:
1549:
1523:
1522:
1503:
1502:
1483:
1482:
1402:
1375:
1374:
1345:
1344:
1308:
1298:
1297:
1272:
1271:
1252:
1251:
1209:
1203:
1169:
1124:
1059:
1035:
1012:
1005:
973:
966:
934:
930:
926:
921:
920:
892:
891:
868:
867:
820:
816:
812:
807:
806:
784:
783:
764:
763:
740:
739:
720:
719:
716:
692:
686:
665:polynomial ring
653:
575:
574:
553:
540:
517:
516:
495:
484:
483:
464:
463:
437:
436:
417:
416:
385:
384:
363:
353:
343:
338:
337:
303:
302:
283:
282:
263:
262:
241:
236:
235:
216:
215:
196:
195:
176:
175:
154:
149:
148:
125:
124:
82:
69:
64:
63:
28:
23:
22:
15:
12:
11:
5:
2730:
2728:
2720:
2719:
2714:
2704:
2703:
2697:
2696:
2694:
2693:
2688:
2683:
2678:
2673:
2668:
2663:
2658:
2652:
2650:
2646:
2645:
2643:
2642:
2637:
2632:
2627:
2622:
2617:
2612:
2607:
2602:
2597:
2591:
2589:
2585:
2584:
2582:
2581:
2576:
2571:
2566:
2565:
2564:
2554:
2553:
2552:
2550:Cubic equation
2542:
2541:
2540:
2530:
2529:
2528:
2518:
2513:
2507:
2505:
2498:
2497:
2488:
2486:
2485:
2478:
2471:
2463:
2455:
2454:
2430:
2423:
2399:
2386:
2385:
2383:
2380:
2379:
2378:
2373:
2368:
2363:
2358:
2353:
2348:
2343:
2336:
2333:
2298:
2295:
2290:
2286:
2269:
2266:
2225:
2221:
2217:
2214:
2194:
2191:
2188:
2185:
2182:
2169:
2166:
2165:
2164:
2153:
2148:
2145:
2142:
2138:
2134:
2129:
2125:
2119:
2115:
2100:
2099:
2088:
2083:
2079:
2064:
2063:
2052:
2047:
2042:
2038:
2032:
2027:
2023:
2017:
2012:
2008:
1993:
1992:
1981:
1978:
1975:
1972:
1969:
1966:
1963:
1960:
1957:
1954:
1951:
1948:
1934:
1933:
1922:
1919:
1916:
1913:
1908:
1904:
1900:
1895:
1891:
1887:
1882:
1878:
1874:
1871:
1868:
1845:
1842:
1839:
1834:
1830:
1826:
1821:
1817:
1813:
1808:
1804:
1785:
1782:
1769:
1749:
1729:
1726:
1723:
1703:
1680:
1675:
1671:
1668:
1665:
1659:
1653:
1647:
1642:
1638:
1635:
1632:
1626:
1600:
1599:
1588:
1580:
1576:
1572:
1569:
1566:
1563:
1559:
1530:
1510:
1490:
1479:Hilbert series
1459:
1456:
1453:
1450:
1447:
1444:
1441:
1438:
1435:
1432:
1427:
1424:
1419:
1413:
1410:
1405:
1401:
1398:
1395:
1392:
1387:
1384:
1358:
1355:
1352:
1326:
1323:
1320:
1317:
1314:
1311:
1307:
1285:
1282:
1279:
1259:
1240:
1239:
1228:
1222:
1218:
1215:
1212:
1206:
1202:
1199:
1196:
1193:
1187:
1184:
1181:
1178:
1175:
1172:
1168:
1163:
1157:
1154:
1151:
1148:
1145:
1142:
1139:
1136:
1133:
1130:
1127:
1122:
1119:
1116:
1113:
1110:
1107:
1104:
1101:
1098:
1095:
1092:
1089:
1086:
1083:
1080:
1077:
1074:
1071:
1068:
1065:
1062:
1056:
1050:
1044:
1041:
1038:
1033:
1030:
1027:
1024:
1021:
1018:
1015:
1009:
1003:
997:
992:
988:
985:
982:
979:
976:
970:
964:
960:
951:
946:
943:
938:
929:
905:
902:
899:
875:
846:
837:
832:
829:
824:
815:
791:
771:
747:
727:
715:
712:
708:monomial basis
690:Monomial basis
688:Main article:
685:
684:Monomial basis
682:
661:monomial basis
652:
649:
626:Puiseux series
622:Laurent series
614:
613:
610:complex number
597:
594:
591:
588:
585:
582:
560:
556:
552:
547:
543:
539:
536:
533:
530:
527:
524:
502:
498:
494:
491:
471:
456:
444:
424:
404:
401:
398:
395:
392:
370:
366:
360:
356:
350:
346:
325:
322:
319:
316:
313:
310:
290:
270:
248:
244:
223:
203:
183:
161:
157:
132:
112:
109:
106:
103:
100:
97:
94:
89:
85:
81:
76:
72:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
2729:
2718:
2715:
2713:
2710:
2709:
2707:
2692:
2691:Gröbner basis
2689:
2687:
2684:
2682:
2679:
2677:
2674:
2672:
2669:
2667:
2664:
2662:
2659:
2657:
2656:Factorization
2654:
2653:
2651:
2647:
2641:
2638:
2636:
2633:
2631:
2628:
2626:
2623:
2621:
2618:
2616:
2613:
2611:
2608:
2606:
2603:
2601:
2598:
2596:
2593:
2592:
2590:
2588:By properties
2586:
2580:
2577:
2575:
2572:
2570:
2567:
2563:
2560:
2559:
2558:
2555:
2551:
2548:
2547:
2546:
2543:
2539:
2536:
2535:
2534:
2531:
2527:
2524:
2523:
2522:
2519:
2517:
2514:
2512:
2509:
2508:
2506:
2504:
2499:
2495:
2491:
2484:
2479:
2477:
2472:
2470:
2465:
2464:
2461:
2450:
2446:
2445:
2440:
2434:
2431:
2426:
2424:0-387-98487-9
2420:
2416:
2412:
2411:
2403:
2400:
2396:
2391:
2388:
2381:
2377:
2374:
2372:
2369:
2367:
2364:
2362:
2359:
2357:
2354:
2352:
2349:
2347:
2344:
2342:
2339:
2338:
2334:
2332:
2330:
2329:
2324:
2320:
2316:
2312:
2296:
2293:
2288:
2284:
2275:
2267:
2265:
2263:
2259:
2258:Gröbner bases
2255:
2251:
2247:
2242:
2239:
2223:
2219:
2215:
2212:
2192:
2189:
2186:
2183:
2180:
2167:
2151:
2146:
2143:
2140:
2136:
2132:
2127:
2123:
2117:
2113:
2105:
2104:
2103:
2086:
2081:
2077:
2069:
2068:
2067:
2050:
2045:
2040:
2036:
2030:
2025:
2021:
2015:
2010:
2006:
1998:
1997:
1996:
1979:
1973:
1970:
1967:
1964:
1961:
1958:
1955:
1949:
1946:
1939:
1938:
1937:
1920:
1914:
1911:
1906:
1902:
1898:
1893:
1889:
1885:
1880:
1876:
1869:
1866:
1859:
1858:
1857:
1856:one can set
1843:
1840:
1837:
1832:
1828:
1824:
1819:
1815:
1811:
1806:
1802:
1793:
1792:
1783:
1781:
1767:
1747:
1727:
1724:
1721:
1701:
1673:
1669:
1666:
1663:
1651:
1640:
1636:
1633:
1630:
1614:variables is
1612:
1606:
1586:
1578:
1570:
1567:
1564:
1557:
1548:
1547:
1546:
1545:expansion of
1544:
1528:
1508:
1488:
1480:
1475:
1473:
1454:
1451:
1448:
1439:
1436:
1433:
1425:
1422:
1417:
1408:
1399:
1396:
1393:
1385:
1382:
1372:
1356:
1353:
1350:
1341:
1324:
1318:
1315:
1312:
1305:
1283:
1280:
1277:
1257:
1249:
1245:
1226:
1216:
1213:
1210:
1200:
1197:
1194:
1185:
1179:
1176:
1173:
1166:
1161:
1152:
1149:
1146:
1140:
1137:
1134:
1131:
1128:
1125:
1117:
1114:
1111:
1108:
1105:
1099:
1096:
1093:
1087:
1084:
1081:
1075:
1069:
1066:
1063:
1054:
1042:
1039:
1036:
1028:
1025:
1022:
1016:
1013:
1001:
990:
986:
983:
980:
977:
974:
962:
958:
944:
941:
927:
919:
918:
917:
903:
900:
897:
889:
886:, or using a
873:
865:
861:
844:
830:
827:
813:
805:
789:
769:
761:
745:
725:
713:
711:
709:
705:
701:
697:
691:
683:
681:
680:
676:
674:
670:
666:
662:
656:
650:
648:
646:
642:
638:
633:
631:
627:
623:
619:
611:
595:
592:
589:
586:
583:
580:
558:
554:
550:
545:
541:
534:
531:
528:
525:
500:
496:
492:
489:
469:
461:
457:
442:
422:
402:
399:
396:
393:
390:
368:
364:
358:
354:
348:
344:
323:
320:
317:
314:
311:
308:
288:
268:
246:
242:
221:
201:
181:
159:
155:
146:
145:empty product
130:
110:
107:
104:
101:
98:
95:
92:
87:
83:
79:
74:
70:
61:
57:
53:
52:power product
49:
48:
47:
45:
41:
37:
33:
19:
2686:Discriminant
2609:
2605:Multivariate
2442:
2433:
2409:
2402:
2394:
2390:
2366:Log-log plot
2326:
2315:group action
2271:
2243:
2240:
2171:
2101:
2065:
1994:
1935:
1789:
1787:
1610:
1604:
1601:
1476:
1370:
1369:) of degree
1342:
1247:
1243:
1241:
717:
707:
704:vector space
693:
678:
677:
672:
657:
654:
634:
615:
51:
35:
29:
2635:Homogeneous
2630:Square-free
2625:Irreducible
2490:Polynomials
460:coefficient
234:or a power
32:mathematics
2706:Categories
2595:Univariate
2439:"Monomial"
2382:References
1270:of degree
40:polynomial
18:Mononomial
2681:Resultant
2620:Trinomial
2600:Bivariate
2449:EMS Press
2371:Power law
2289:α
2147:β
2141:α
2128:β
2118:α
2082:α
2051:⋯
1974:…
1947:α
1915:…
1841:…
1568:−
1412:¯
1316:−
1281:−
1221:¯
1214:−
1177:−
1150:−
1141:×
1138:⋯
1135:×
1129:×
1115:−
1100:×
1097:⋯
1094:×
1076:×
1040:−
1026:−
984:−
645:haplology
529:−
490:−
56:variables
2666:Division
2615:Binomial
2610:Monomial
2335:See also
2268:Geometry
673:monomial
36:monomial
2717:Algebra
2451:, 2001
2397:, 1969.
1541:of the
862:, as a
702:of the
667:, or a
641:syncope
281:, with
147:and to
2503:degree
2421:
2317:of an
2168:Degree
714:Number
637:prefix
700:basis
663:of a
383:with
58:with
2492:and
2419:ISBN
1936:and
1788:The
1477:The
620:and
515:and
44:term
34:, a
2501:By
2272:In
1760:in
1714:in
1608:in
1501:in
1373:is
890:of
866:in
762:of
738:in
643:by
261:of
30:In
2708::
2447:,
2441:,
2417:.
2331:.
2264:.
1474:.
1340:.
916::
632:.
612:).
559:13
455:).
2482:e
2475:t
2468:v
2427:.
2415:1
2297:0
2294:=
2285:x
2224:2
2220:z
2216:y
2213:x
2193:c
2190:+
2187:b
2184:+
2181:a
2152:.
2144:+
2137:x
2133:=
2124:x
2114:x
2087:.
2078:x
2046:c
2041:3
2037:x
2031:b
2026:2
2022:x
2016:a
2011:1
2007:x
1980:.
1977:)
1971:,
1968:c
1965:,
1962:b
1959:,
1956:a
1953:(
1950:=
1921:,
1918:)
1912:,
1907:3
1903:x
1899:,
1894:2
1890:x
1886:,
1881:1
1877:x
1873:(
1870:=
1867:x
1844:,
1838:,
1833:3
1829:x
1825:,
1820:2
1816:x
1812:,
1807:1
1803:x
1768:n
1748:d
1728:1
1725:+
1722:n
1702:d
1679:)
1674:d
1670:d
1667:+
1664:n
1658:(
1652:=
1646:)
1641:n
1637:d
1634:+
1631:n
1625:(
1611:n
1605:d
1587:.
1579:n
1575:)
1571:t
1565:1
1562:(
1558:1
1529:d
1509:n
1489:d
1458:)
1455:2
1452:+
1449:d
1446:(
1443:)
1440:1
1437:+
1434:d
1431:(
1426:2
1423:1
1418:=
1409:2
1404:)
1400:1
1397:+
1394:d
1391:(
1386:2
1383:1
1371:d
1357:3
1354:=
1351:n
1325:!
1322:)
1319:1
1313:n
1310:(
1306:1
1284:1
1278:n
1258:d
1248:d
1244:n
1227:.
1217:1
1211:n
1205:)
1201:1
1198:+
1195:d
1192:(
1186:!
1183:)
1180:1
1174:n
1171:(
1167:1
1162:=
1156:)
1153:1
1147:n
1144:(
1132:2
1126:1
1121:)
1118:1
1112:n
1109:+
1106:d
1103:(
1091:)
1088:2
1085:+
1082:d
1079:(
1073:)
1070:1
1067:+
1064:d
1061:(
1055:=
1049:)
1043:1
1037:n
1032:)
1029:1
1023:n
1020:(
1017:+
1014:d
1008:(
1002:=
996:)
991:d
987:1
981:d
978:+
975:n
969:(
963:=
959:)
950:)
945:d
942:n
937:(
928:(
904:1
901:+
898:d
874:d
845:)
836:)
831:d
828:n
823:(
814:(
790:n
770:d
746:n
726:d
596:,
593:z
590:,
587:y
584:,
581:x
555:z
551:y
546:4
542:x
538:)
535:i
532:4
526:3
523:(
501:5
497:x
493:7
470:1
443:1
423:0
403:c
400:,
397:b
394:,
391:a
369:c
365:z
359:b
355:y
349:a
345:x
324:,
321:z
318:,
315:y
312:,
309:x
289:n
269:x
247:n
243:x
222:1
202:x
182:x
160:0
156:x
131:1
111:z
108:z
105:z
102:y
99:x
96:x
93:=
88:3
84:z
80:y
75:2
71:x
20:)
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