25:
1454:
1221:
803:
487:
946:
382:
654:
606:
552:
266:
1082:
710:
1517:
995:
1274:
997:
form a vector space (if the coefficients belong to a field) or a free module (if the coefficients belong to a ring), which has the set of all monomials as a basis, called the
875:
1326:
1543:
1578:
830:
1321:
1301:
1106:
1026:
1111:
719:
394:
304:
611:
563:
496:
213:
97:
69:
880:
1476:
76:
116:
877:
an exponent equal to zero means that the corresponding indeterminate does not appear in the monomial; in particular
54:
46:
83:
50:
1522:
1462:
of the monomial basis in the multivariate case. For problems which require choosing a total order, such as
1035:
65:
666:
954:
1226:
1085:
35:
839:
39:
1005:
292:
142:
301:
form also a vector space (or a free module in the case of a ring of coefficients), which has
195:
1548:
90:
1638:
1277:
808:
8:
1608:
1603:
1593:
154:
1588:
1306:
1286:
1091:
1011:
275:
174:
158:
1323:
as a basis. The number of these monomials is the dimension of this subspace, equal to
1598:
1449:{\displaystyle {\binom {d+n}{d}}={\binom {d+n}{n}}={\frac {(d+1)\cdots (d+n)}{n!}}.}
1633:
177:
of monomials (this is an immediate consequence of the definition of a polynomial).
1463:
1029:
186:
138:
1618:
1613:
1470:
490:
1627:
146:
1459:
557:
285:
162:
150:
130:
1216:{\displaystyle {\binom {d+n-1}{d}}={\frac {n(n+1)\cdots (n+d-1)}{d!}},}
170:
24:
713:
166:
951:
Similar to the case of univariate polynomials, the polynomials in
833:
1303:
form also a subspace, which has the monomials of degree at most
798:{\displaystyle x_{1}^{d_{1}}x_{2}^{d_{2}}\cdots x_{n}^{d_{n}},}
1473:– that is, a total order on the set of monomials such that
482:{\displaystyle a_{0}+a_{1}x+a_{2}x^{2}+\dots +a_{d}x^{d},}
1458:
In contrast to the univariate case, there is no natural
1229:
1088:
of this subspace is the number of monomials of degree
1551:
1525:
1479:
1329:
1309:
1289:
1114:
1094:
1038:
1014:
957:
883:
842:
811:
722:
669:
614:
566:
499:
397:
307:
216:
941:{\displaystyle 1=x_{1}^{0}x_{2}^{0}\cdots x_{n}^{0}}
377:{\displaystyle \{1,x,x^{2},\ldots ,x^{d-1},x^{d}\}}
1572:
1537:
1511:
1448:
1315:
1295:
1268:
1215:
1100:
1076:
1020:
989:
940:
869:
824:
797:
704:
648:
600:
546:
481:
376:
260:
1387:
1366:
1354:
1333:
1145:
1118:
391:of a polynomial is its expression on this basis:
1625:
1260:
1233:
371:
308:
16:Basis of polynomials consisting of monomials
649:{\displaystyle 1>x>x^{2}>\cdots .}
601:{\displaystyle 1<x<x^{2}<\cdots ,}
268:as an (infinite) basis. More generally, if
169:. The monomials form a basis because every
53:. Unsourced material may be challenged and
1493:
1489:
547:{\displaystyle \sum _{i=0}^{d}a_{i}x^{i}.}
658:
117:Learn how and when to remove this message
1466:computations, one generally chooses an
261:{\displaystyle 1,x,x^{2},x^{3},\ldots }
1626:
663:In the case of several indeterminates
1077:{\displaystyle d=d_{1}+\cdots +d_{n}}
705:{\displaystyle x_{1},\ldots ,x_{n},}
180:
173:may be uniquely written as a finite
51:adding citations to reliable sources
18:
1512:{\displaystyle m<n\iff mq<nq}
990:{\displaystyle x_{1},\ldots ,x_{n}}
13:
1370:
1337:
1283:The polynomials of degree at most
1237:
1122:
1032:which has the monomials of degree
14:
1650:
1269:{\textstyle {\binom {d+n-1}{d}}}
556:The monomial basis is naturally
23:
560:, either by increasing degrees
1490:
1429:
1417:
1411:
1399:
1196:
1178:
1172:
1160:
1:
870:{\displaystyle x_{i}^{0}=1,}
7:
1582:
10:
1655:
288:which has the same basis.
210:-vector space, which has
608:or by decreasing degrees
1538:{\displaystyle 1\leq m}
1006:homogeneous polynomials
165:) that consists of all
1574:
1573:{\displaystyle m,n,q.}
1539:
1513:
1450:
1317:
1297:
1270:
1217:
1102:
1078:
1022:
991:
942:
871:
826:
799:
706:
659:Several indeterminates
650:
602:
548:
520:
489:or, using the shorter
483:
378:
262:
196:univariate polynomials
1575:
1540:
1514:
1451:
1318:
1298:
1271:
1218:
1103:
1079:
1023:
992:
943:
872:
827:
825:{\displaystyle d_{i}}
800:
707:
651:
603:
549:
500:
484:
379:
263:
1549:
1523:
1477:
1327:
1307:
1287:
1278:binomial coefficient
1227:
1112:
1092:
1036:
1012:
955:
881:
840:
809:
720:
667:
612:
564:
497:
395:
305:
214:
47:improve this article
1609:Legendre polynomial
1604:Lagrange polynomial
1594:Polynomial sequence
1545:for every monomial
937:
919:
904:
857:
791:
766:
744:
291:The polynomials of
1570:
1535:
1509:
1446:
1313:
1293:
1266:
1213:
1098:
1074:
1018:
987:
938:
923:
905:
890:
867:
843:
822:
795:
770:
745:
723:
702:
646:
598:
544:
479:
374:
258:
175:linear combination
1599:Newton polynomial
1441:
1385:
1352:
1316:{\displaystyle d}
1296:{\displaystyle d}
1258:
1208:
1143:
1101:{\displaystyle d}
1021:{\displaystyle d}
832:are non-negative
181:One indeterminate
127:
126:
119:
101:
1646:
1579:
1577:
1576:
1571:
1544:
1542:
1541:
1536:
1518:
1516:
1515:
1510:
1455:
1453:
1452:
1447:
1442:
1440:
1432:
1397:
1392:
1391:
1390:
1381:
1369:
1359:
1358:
1357:
1348:
1336:
1322:
1320:
1319:
1314:
1302:
1300:
1299:
1294:
1275:
1273:
1272:
1267:
1265:
1264:
1263:
1254:
1236:
1222:
1220:
1219:
1214:
1209:
1207:
1199:
1155:
1150:
1149:
1148:
1139:
1121:
1107:
1105:
1104:
1099:
1084:as a basis. The
1083:
1081:
1080:
1075:
1073:
1072:
1054:
1053:
1027:
1025:
1024:
1019:
996:
994:
993:
988:
986:
985:
967:
966:
947:
945:
944:
939:
936:
931:
918:
913:
903:
898:
876:
874:
873:
868:
856:
851:
831:
829:
828:
823:
821:
820:
804:
802:
801:
796:
790:
789:
788:
778:
765:
764:
763:
753:
743:
742:
741:
731:
711:
709:
708:
703:
698:
697:
679:
678:
655:
653:
652:
647:
636:
635:
607:
605:
604:
599:
588:
587:
553:
551:
550:
545:
540:
539:
530:
529:
519:
514:
488:
486:
485:
480:
475:
474:
465:
464:
446:
445:
436:
435:
420:
419:
407:
406:
383:
381:
380:
375:
370:
369:
357:
356:
332:
331:
300:
283:
273:
267:
265:
264:
259:
251:
250:
238:
237:
209:
203:
193:
122:
115:
111:
108:
102:
100:
66:"Monomial basis"
59:
27:
19:
1654:
1653:
1649:
1648:
1647:
1645:
1644:
1643:
1624:
1623:
1589:Horner's method
1585:
1550:
1547:
1546:
1524:
1521:
1520:
1478:
1475:
1474:
1433:
1398:
1396:
1386:
1371:
1365:
1364:
1363:
1353:
1338:
1332:
1331:
1330:
1328:
1325:
1324:
1308:
1305:
1304:
1288:
1285:
1284:
1259:
1238:
1232:
1231:
1230:
1228:
1225:
1224:
1200:
1156:
1154:
1144:
1123:
1117:
1116:
1115:
1113:
1110:
1109:
1093:
1090:
1089:
1068:
1064:
1049:
1045:
1037:
1034:
1033:
1013:
1010:
1009:
981:
977:
962:
958:
956:
953:
952:
948:is a monomial.
932:
927:
914:
909:
899:
894:
882:
879:
878:
852:
847:
841:
838:
837:
816:
812:
810:
807:
806:
784:
780:
779:
774:
759:
755:
754:
749:
737:
733:
732:
727:
721:
718:
717:
693:
689:
674:
670:
668:
665:
664:
661:
631:
627:
613:
610:
609:
583:
579:
565:
562:
561:
558:totally ordered
535:
531:
525:
521:
515:
504:
498:
495:
494:
470:
466:
460:
456:
441:
437:
431:
427:
415:
411:
402:
398:
396:
393:
392:
365:
361:
346:
342:
327:
323:
306:
303:
302:
296:
279:
269:
246:
242:
233:
229:
215:
212:
211:
205:
199:
189:
187:polynomial ring
183:
139:polynomial ring
123:
112:
106:
103:
60:
58:
44:
28:
17:
12:
11:
5:
1652:
1642:
1641:
1636:
1622:
1621:
1619:Chebyshev form
1616:
1614:Bernstein form
1611:
1606:
1601:
1596:
1591:
1584:
1581:
1569:
1566:
1563:
1560:
1557:
1554:
1534:
1531:
1528:
1508:
1505:
1502:
1499:
1496:
1492:
1488:
1485:
1482:
1471:monomial order
1445:
1439:
1436:
1431:
1428:
1425:
1422:
1419:
1416:
1413:
1410:
1407:
1404:
1401:
1395:
1389:
1384:
1380:
1377:
1374:
1368:
1362:
1356:
1351:
1347:
1344:
1341:
1335:
1312:
1292:
1262:
1257:
1253:
1250:
1247:
1244:
1241:
1235:
1212:
1206:
1203:
1198:
1195:
1192:
1189:
1186:
1183:
1180:
1177:
1174:
1171:
1168:
1165:
1162:
1159:
1153:
1147:
1142:
1138:
1135:
1132:
1129:
1126:
1120:
1097:
1071:
1067:
1063:
1060:
1057:
1052:
1048:
1044:
1041:
1017:
999:monomial basis
984:
980:
976:
973:
970:
965:
961:
935:
930:
926:
922:
917:
912:
908:
902:
897:
893:
889:
886:
866:
863:
860:
855:
850:
846:
819:
815:
794:
787:
783:
777:
773:
769:
762:
758:
752:
748:
740:
736:
730:
726:
701:
696:
692:
688:
685:
682:
677:
673:
660:
657:
645:
642:
639:
634:
630:
626:
623:
620:
617:
597:
594:
591:
586:
582:
578:
575:
572:
569:
543:
538:
534:
528:
524:
518:
513:
510:
507:
503:
491:sigma notation
478:
473:
469:
463:
459:
455:
452:
449:
444:
440:
434:
430:
426:
423:
418:
414:
410:
405:
401:
389:canonical form
373:
368:
364:
360:
355:
352:
349:
345:
341:
338:
335:
330:
326:
322:
319:
316:
313:
310:
257:
254:
249:
245:
241:
236:
232:
228:
225:
222:
219:
182:
179:
135:monomial basis
125:
124:
31:
29:
22:
15:
9:
6:
4:
3:
2:
1651:
1640:
1637:
1635:
1632:
1631:
1629:
1620:
1617:
1615:
1612:
1610:
1607:
1605:
1602:
1600:
1597:
1595:
1592:
1590:
1587:
1586:
1580:
1567:
1564:
1561:
1558:
1555:
1552:
1532:
1529:
1526:
1506:
1503:
1500:
1497:
1494:
1486:
1483:
1480:
1472:
1469:
1465:
1464:Gröbner basis
1461:
1456:
1443:
1437:
1434:
1426:
1423:
1420:
1414:
1408:
1405:
1402:
1393:
1382:
1378:
1375:
1372:
1360:
1349:
1345:
1342:
1339:
1310:
1290:
1281:
1279:
1255:
1251:
1248:
1245:
1242:
1239:
1210:
1204:
1201:
1193:
1190:
1187:
1184:
1181:
1175:
1169:
1166:
1163:
1157:
1151:
1140:
1136:
1133:
1130:
1127:
1124:
1095:
1087:
1069:
1065:
1061:
1058:
1055:
1050:
1046:
1042:
1039:
1031:
1015:
1007:
1002:
1000:
982:
978:
974:
971:
968:
963:
959:
949:
933:
928:
924:
920:
915:
910:
906:
900:
895:
891:
887:
884:
864:
861:
858:
853:
848:
844:
835:
817:
813:
792:
785:
781:
775:
771:
767:
760:
756:
750:
746:
738:
734:
728:
724:
716:is a product
715:
699:
694:
690:
686:
683:
680:
675:
671:
656:
643:
640:
637:
632:
628:
624:
621:
618:
615:
595:
592:
589:
584:
580:
576:
573:
570:
567:
559:
554:
541:
536:
532:
526:
522:
516:
511:
508:
505:
501:
492:
476:
471:
467:
461:
457:
453:
450:
447:
442:
438:
432:
428:
424:
421:
416:
412:
408:
403:
399:
390:
385:
366:
362:
358:
353:
350:
347:
343:
339:
336:
333:
328:
324:
320:
317:
314:
311:
299:
294:
289:
287:
282:
277:
272:
255:
252:
247:
243:
239:
234:
230:
226:
223:
220:
217:
208:
202:
198:over a field
197:
192:
188:
178:
176:
172:
168:
164:
160:
156:
152:
148:
144:
140:
136:
132:
121:
118:
110:
99:
96:
92:
89:
85:
82:
78:
75:
71:
68: –
67:
63:
62:Find sources:
56:
52:
48:
42:
41:
37:
32:This article
30:
26:
21:
20:
1467:
1457:
1282:
1108:, which is
1003:
998:
950:
662:
555:
388:
386:
384:as a basis.
297:
290:
280:
270:
206:
200:
190:
184:
163:coefficients
147:vector space
134:
128:
113:
104:
94:
87:
80:
73:
61:
45:Please help
33:
1639:Polynomials
1460:total order
286:free module
151:free module
131:mathematics
1628:Categories
1468:admissible
1008:of degree
805:where the
171:polynomial
77:newspapers
1530:≤
1491:⟺
1415:⋯
1249:−
1191:−
1176:⋯
1134:−
1086:dimension
1059:⋯
972:…
921:⋯
768:⋯
684:…
641:⋯
593:⋯
502:∑
451:⋯
351:−
337:…
256:…
167:monomials
153:over the
34:does not
1583:See also
1030:subspace
834:integers
714:monomial
295:at most
107:May 2022
1634:Algebra
1028:form a
141:is its
91:scholar
55:removed
40:sources
1223:where
293:degree
145:(as a
93:
86:
79:
72:
64:
1276:is a
836:. As
284:is a
278:then
274:is a
204:is a
155:field
143:basis
137:of a
98:JSTOR
84:books
1519:and
1501:<
1484:<
1004:The
638:>
625:>
619:>
590:<
577:<
571:<
387:The
276:ring
185:The
159:ring
133:the
70:news
38:any
36:cite
194:of
161:of
157:or
149:or
129:In
49:by
1630::
1280:.
1001:.
712:a
493::
1568:.
1565:q
1562:,
1559:n
1556:,
1553:m
1533:m
1527:1
1507:q
1504:n
1498:q
1495:m
1487:n
1481:m
1444:.
1438:!
1435:n
1430:)
1427:n
1424:+
1421:d
1418:(
1412:)
1409:1
1406:+
1403:d
1400:(
1394:=
1388:)
1383:n
1379:n
1376:+
1373:d
1367:(
1361:=
1355:)
1350:d
1346:n
1343:+
1340:d
1334:(
1311:d
1291:d
1261:)
1256:d
1252:1
1246:n
1243:+
1240:d
1234:(
1211:,
1205:!
1202:d
1197:)
1194:1
1188:d
1185:+
1182:n
1179:(
1173:)
1170:1
1167:+
1164:n
1161:(
1158:n
1152:=
1146:)
1141:d
1137:1
1131:n
1128:+
1125:d
1119:(
1096:d
1070:n
1066:d
1062:+
1056:+
1051:1
1047:d
1043:=
1040:d
1016:d
983:n
979:x
975:,
969:,
964:1
960:x
934:0
929:n
925:x
916:0
911:2
907:x
901:0
896:1
892:x
888:=
885:1
865:,
862:1
859:=
854:0
849:i
845:x
818:i
814:d
793:,
786:n
782:d
776:n
772:x
761:2
757:d
751:2
747:x
739:1
735:d
729:1
725:x
700:,
695:n
691:x
687:,
681:,
676:1
672:x
644:.
633:2
629:x
622:x
616:1
596:,
585:2
581:x
574:x
568:1
542:.
537:i
533:x
527:i
523:a
517:d
512:0
509:=
506:i
477:,
472:d
468:x
462:d
458:a
454:+
448:+
443:2
439:x
433:2
429:a
425:+
422:x
417:1
413:a
409:+
404:0
400:a
372:}
367:d
363:x
359:,
354:1
348:d
344:x
340:,
334:,
329:2
325:x
321:,
318:x
315:,
312:1
309:{
298:d
281:K
271:K
253:,
248:3
244:x
240:,
235:2
231:x
227:,
224:x
221:,
218:1
207:K
201:K
191:K
120:)
114:(
109:)
105:(
95:·
88:·
81:·
74:·
57:.
43:.
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