28:
1692:
515:
of finite rank; it suffices to replace the dimension by the rank. Often the graded vector space or module of which the
Hilbert–Poincaré series is considered has additional structure, for instance, that of a ring, but the Hilbert–Poincaré series is independent of the multiplicative or other structure.
233:
in cases when the latter exists; however, the
Hilbert–Poincaré series describes the rank in every degree, while the Hilbert polynomial describes it only in all but finitely many degrees, and therefore provides less information. In particular the Hilbert–Poincaré series cannot be deduced from the
1527:
1165:
1687:{\displaystyle 0\to C^{0}{\stackrel {d_{0}}{\longrightarrow }}C^{1}{\stackrel {d_{1}}{\longrightarrow }}C^{2}{\stackrel {d_{2}}{\longrightarrow }}\cdots {\stackrel {d_{n-1}}{\longrightarrow }}C^{n}\longrightarrow 0.}
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Hilbert polynomial even if the latter exists. In good cases, the
Hilbert–Poincaré series can be expressed as a
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1700:
841:
896:
742:
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522:
1010:
1697:
The
Hilbert–Poincaré series (here often called the Poincaré polynomial) of the graded vector space
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477:
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algebraic structures (where the dimension of the entire structure is often infinite). It is a
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gives the dimension (or rank) of the sub-structure of elements homogeneous of degree
143:
616:(by induction, say), one can deduce that the sum of the Hilbert–Poincaré series of
1171:
Since the length is additive, Poincaré series are also additive. Hence, we have:
512:
125:
27:
1836:
1160:{\displaystyle 0\to K(-d_{n})\to M(-d_{n}){\overset {x_{n}}{\to }}M\to C\to 0}
151:
129:
507:
in which each submodule of elements homogeneous of a fixed degree
1952:
A famous relation between the two is that there is a polynomial
316:{\displaystyle V=\textstyle \bigoplus _{i\in \mathbb {N} }V_{i}}
464:{\displaystyle \sum _{i\in \mathbb {N} }\dim _{K}(V_{i})t^{i}.}
1505:. The theorem thus now follows from the inductive hypothesis.
21:
1306:{\displaystyle P(M,t)=-P(K(-d_{n}),t)+P(M(-d_{n}),t)-P(C,t)}
384:
is finite-dimensional. Then the
Hilbert–Poincaré series of
1942:{\displaystyle P_{H}(t)=\sum _{j=0}^{n}\dim(H^{j})t^{j}.}
1825:{\displaystyle P_{C}(t)=\sum _{j=0}^{n}\dim(C^{j})t^{j}.}
977:
is large enough. Next, suppose the theorem is true for
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is a polynomial with integral coefficients divided by
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1987:
1958:
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1743:
1703:
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1513:
An example of graded vector space is associated to a
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52:. Unsourced material may be challenged and removed.
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2102:, Ch. 11, an example just after Proposition 11.3.
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2087:
1402:{\displaystyle P(M(-d_{n}),t)=t^{d_{n}}P(M,t)}
887:. The standard proof today is an induction on
903:), which gives more homological information.
8:
2065:{\displaystyle P_{C}(t)-P_{H}(t)=(1+t)Q(t).}
1521:of vector spaces; the latter takes the form
1440:, we can regard it as a graded module over
906:Here is a proof by induction on the number
739:is a finitely generated graded module over
1981:with non-negative coefficients, such that
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891:. Hilbert's original proof made a use of
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112:Learn how and when to remove this message
2080:
1835:The Hilbert–Poincaré polynomial of the
1007:(exact degree-wise), with the notation
18:Hilbert series and Hilbert polynomial
7:
150:, is an adaptation of the notion of
128:, and in particular in the field of
50:adding citations to reliable sources
2132:Introduction to Commutative Algebra
1727:{\displaystyle \bigoplus _{i}C^{i}}
1003:and consider the exact sequence of
880:{\displaystyle \prod (1-t^{d_{i}})}
819:{\displaystyle A,\deg x_{i}=d_{i}}
609:{\displaystyle X_{0},\dots ,X_{n}}
533:
14:
559:{\displaystyle {\binom {n+k}{n}}}
1055:{\displaystyle N(l)_{k}=N_{k+l}}
229:. It is closely related to the
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37:needs additional citations for
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834:. Then the Poincaré series of
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1:
720:{\displaystyle 1/(1-t)^{n+1}}
489:{\displaystyle \mathbb {N} }
338:{\displaystyle \mathbb {N} }
2112:Atiyah & Macdonald 1969
2100:Atiyah & Macdonald 1969
2088:Atiyah & Macdonald 1969
182:, where the coefficient of
136:(also known under the name
2200:
162:in one indeterminate, say
15:
1839:, with cohomology spaces
519:Example: Since there are
61:"Hilbert–Poincaré series"
893:Hilbert's syzygy theorem
2124:Atiyah, Michael Francis
2114:, Ch. 11, Theorem 11.1.
1501:; the same is true for
966:{\displaystyle M_{k}=0}
134:Hilbert–Poincaré series
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1433:{\displaystyle x_{n}}
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731:Hilbert–Serre theorem
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373:{\displaystyle V_{i}}
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202:{\displaystyle t^{n}}
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1974:{\displaystyle Q(t)}
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46:improve this article
2174:Mathematical series
2169:Commutative algebra
2159:Homological algebra
996:{\displaystyle n-1}
940:has finite length,
929:{\displaystyle n=0}
390:formal power series
347:graded vector space
160:formal power series
2134:. Westview Press.
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1494:{\displaystyle A}
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673:rational function
664:{\displaystyle K}
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500:-module over any
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251:{\displaystyle t}
236:rational function
222:{\displaystyle n}
175:{\displaystyle t}
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2128:Macdonald, I.G.
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142:), named after
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11:
5:
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2179:Henri Poincaré
2176:
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2164:Linear algebra
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1005:graded modules
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936:, then, since
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148:Henri Poincaré
139:Hilbert series
120:
119:
34:
32:
25:
13:
10:
9:
6:
4:
3:
2:
2196:
2185:
2184:David Hilbert
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1515:chain complex
1509:Chain complex
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1480:
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1421:
1413:is killed by
1412:
1393:
1390:
1387:
1381:
1374:
1370:
1365:
1361:
1355:
1352:
1344:
1340:
1336:
1330:
1324:
1317:We can write
1297:
1294:
1291:
1285:
1282:
1276:
1273:
1265:
1261:
1257:
1251:
1245:
1242:
1236:
1233:
1225:
1221:
1217:
1211:
1205:
1202:
1199:
1193:
1190:
1187:
1181:
1174:
1173:
1172:
1154:
1148:
1142:
1135:
1131:
1117:
1113:
1109:
1103:
1092:
1088:
1084:
1078:
1072:
1065:
1064:
1063:
1047:
1044:
1041:
1037:
1033:
1028:
1020:
1014:
1006:
990:
987:
984:
976:
960:
957:
952:
948:
939:
923:
920:
917:
909:
904:
902:
898:
894:
890:
867:
863:
858:
854:
851:
845:
837:
833:
829:
828:Artinian ring
811:
807:
803:
798:
794:
790:
787:
784:
776:
772:
768:
765:
762:
757:
753:
746:
738:
730:
728:
712:
709:
706:
698:
695:
692:
685:
681:
674:
653:
649:
645:
642:
639:
634:
630:
623:
601:
597:
593:
590:
587:
582:
578:
570:in variables
569:
545:
541:
538:
535:
517:
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510:
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503:
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458:
453:
449:
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436:
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409:
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402:
394:
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391:
387:
383:
365:
361:
352:
348:
307:
303:
292:
289:
285:
280:
277:
269:
261:
259:
245:
237:
232:
216:
194:
190:
169:
161:
157:
153:
149:
145:
144:David Hilbert
141:
140:
135:
131:
127:
116:
113:
105:
94:
91:
87:
84:
80:
77:
73:
70:
66:
63: –
62:
58:
57:Find sources:
51:
47:
41:
40:
35:This article
33:
29:
24:
23:
19:
2131:
2107:
2095:
2083:
1951:
1848:
1844:
1840:
1834:
1696:
1518:
1512:
1502:
1410:
1316:
1170:
974:
937:
907:
905:
900:
888:
835:
831:
736:
734:
567:
518:
508:
504:
497:
473:
385:
381:
350:
267:
265:
137:
133:
123:
108:
99:
89:
82:
75:
68:
56:
44:Please help
39:verification
36:
126:mathematics
102:August 2020
2153:Categories
2075:References
1837:cohomology
262:Definition
72:newspapers
16:See also:
2090:, Ch. 11.
2008:−
1908:
1885:∑
1791:
1768:∑
1706:⨁
1679:⟶
1657:−
1644:⟶
1637:⋯
1618:⟶
1585:⟶
1552:⟶
1535:→
1481:−
1467:…
1337:−
1283:−
1258:−
1218:−
1203:−
1152:→
1146:→
1128:→
1110:−
1101:→
1085:−
1076:→
988:−
855:−
846:∏
791:
766:…
696:−
643:…
591:…
430:
410:∈
403:∑
293:∈
286:⨁
152:dimension
2130:(1969).
1409:. Since
826:with an
735:Suppose
496:-graded
1851:), is
671:is the
388:is the
130:algebra
86:scholar
2138:
323:be an
156:graded
88:
81:
74:
67:
59:
349:over
93:JSTOR
79:books
2136:ISBN
513:free
266:Let
146:and
132:, a
65:news
1905:dim
1788:dim
973:if
899:of
895:(a
788:deg
511:is
421:dim
124:In
48:by
2155::
2126:;
1682:0.
1062:,
727:.
258:.
2144:.
2060:.
2057:)
2054:t
2051:(
2048:Q
2045:)
2042:t
2039:+
2036:1
2033:(
2030:=
2027:)
2024:t
2021:(
2016:H
2012:P
2005:)
2002:t
1999:(
1994:C
1990:P
1969:)
1966:t
1963:(
1960:Q
1937:.
1932:j
1928:t
1924:)
1919:j
1915:H
1911:(
1900:n
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1892:=
1889:j
1881:=
1878:)
1875:t
1872:(
1867:H
1863:P
1849:C
1847:(
1845:H
1841:H
1820:.
1815:j
1811:t
1807:)
1802:j
1798:C
1794:(
1783:n
1778:0
1775:=
1772:j
1764:=
1761:)
1758:t
1755:(
1750:C
1746:P
1720:i
1716:C
1710:i
1674:n
1670:C
1660:1
1654:n
1650:d
1628:2
1624:d
1609:2
1605:C
1595:1
1591:d
1576:1
1572:C
1562:0
1558:d
1543:0
1539:C
1532:0
1519:C
1503:C
1489:]
1484:1
1478:n
1474:x
1470:,
1464:,
1459:0
1455:x
1451:[
1448:A
1426:n
1422:x
1411:K
1397:)
1394:t
1391:,
1388:M
1385:(
1382:P
1375:n
1371:d
1366:t
1362:=
1359:)
1356:t
1353:,
1350:)
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1341:d
1334:(
1331:M
1328:(
1325:P
1313:.
1301:)
1298:t
1295:,
1292:C
1289:(
1286:P
1280:)
1277:t
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1271:)
1266:n
1262:d
1255:(
1252:M
1249:(
1246:P
1243:+
1240:)
1237:t
1234:,
1231:)
1226:n
1222:d
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1212:K
1209:(
1206:P
1200:=
1197:)
1194:t
1191:,
1188:M
1185:(
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1167:.
1155:0
1149:C
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1136:n
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1123:)
1118:n
1114:d
1107:(
1104:M
1098:)
1093:n
1089:d
1082:(
1079:K
1073:0
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1042:k
1038:N
1034:=
1029:k
1025:)
1021:l
1018:(
1015:N
991:1
985:n
975:k
961:0
958:=
953:k
949:M
938:M
924:0
921:=
918:n
908:n
901:M
889:n
875:)
868:i
864:d
859:t
852:1
849:(
836:M
832:A
812:i
808:d
804:=
799:i
795:x
785:,
782:]
777:n
773:x
769:,
763:,
758:1
754:x
750:[
747:A
737:M
713:1
710:+
707:n
703:)
699:t
693:1
690:(
686:/
682:1
659:]
654:n
650:X
646:,
640:,
635:0
631:X
627:[
624:K
602:n
598:X
594:,
588:,
583:0
579:X
568:k
551:)
546:n
542:k
539:+
536:n
530:(
509:n
505:R
498:R
483:N
459:.
454:i
450:t
446:)
441:i
437:V
433:(
425:K
414:N
407:i
386:V
382:i
366:i
362:V
351:K
345:-
332:N
308:i
304:V
297:N
290:i
281:=
278:V
268:K
246:t
217:n
195:n
191:t
170:t
115:)
109:(
104:)
100:(
90:·
83:·
76:·
69:·
42:.
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