25:
1956:
1236:
The second step, which is usually the fastest, may be accelerated in the following way: Firstly, the Gröbner basis is replaced by the list of its leading monomials (this is already done for the computation of the
Hilbert series). Then each monomial like
862:. Moreover, the dimension is not changed if the polynomials of the Gröbner basis are replaced with their leading monomials, and if these leading monomials are replaced with their radical (monomials obtained by removing exponents). So:
1821:
Cox, David A.; Little, John; O'Shea, Donal Ideals, varieties, and algorithms. An introduction to computational algebraic geometry and commutative algebra. Fourth edition. Undergraduate Texts in
Mathematics. Springer, Cham,
740:
has a constant dimension. This can also be deduced from the result stated below the third definition, and the fact that the dimension of the tangent space is equal to the Krull dimension at any non-singular point (see
1404:
583:
481:
1696:(that is defined by polynomials with real coefficients), it may occur that the real dimension of the set of its real points is smaller than its dimension as a semi algebraic set. For example, the
1307:
1757:
710:
if defined over the reals, then the set of its real regular points, if it is not empty, is a differentiable manifold that has the same dimension as a variety and as a manifold.
1229:
Without further information on the system, there is only one practical method, which consists of computing a Gröbner basis and deducing the degree of the denominator of the
322:
1575:
288:
1137:
896:
1682:
1662:
1638:
1618:
1595:
1532:
1512:
1489:
1224:
992:
972:
944:
916:
655:
633:
422:
42:
89:
61:
1766:(that is the set of real solutions of a single polynomial equation), there exists a probabilistic algorithm to compute its real dimension.
68:
1312:
1887:
529:
427:
771:
This definition is not intrinsic as it apply only to algebraic sets that are explicitly embedded in an affine or projective space.
75:
724:
688:
57:
1809:
1240:
1759:
is an algebraic variety of dimension two, which has only one real point (0, 0, 0), and thus has the real dimension zero.
1182:
appear explicitly in the definition, the value of the dimension must be reduced by one. For example, the dimension of
1990:
108:
1804:
Chapter 11 of Atiyah, Michael
Francis; Macdonald, I.G. (1969), Introduction to Commutative Algebra, Westview Press,
1880:
1203:
859:
189:
82:
46:
371:). The dimension is also independent of the choice of coordinates; in other words it does not change if the
2179:
1975:
1703:
1833:
1780:
2189:
1873:
1762:
The real dimension is more difficult to compute than the algebraic dimension. For the case of a real
140:
Some of these definitions are of geometric nature, while some other are purely algebraic and rely on
1910:
1775:
1056:
868:
501:. It is thus probably the definition that gives the easiest intuitive description of the notion.
2118:
2113:
2093:
1418:
1410:
of the variables, such that none of these products of variables depends only on the variables in
701:
35:
2103:
2098:
2078:
1038:
2108:
2088:
2083:
1785:
742:
1858:, Proceedings of the 2015 international symposium on Symbolic and algebraic computation, ACM
1017:
383:
325:
293:
196:
1541:
229:
8:
1985:
1980:
1853:
1685:
1068:
919:
873:
523:
368:
175:
141:
2184:
2159:
2000:
1955:
1667:
1647:
1623:
1603:
1580:
1517:
1497:
1474:
1209:
1025:
1000:
977:
957:
929:
901:
829:
640:
618:
407:
386:
205:
126:
1641:
1995:
1805:
1697:
1456:
1446:
1226:, it may be difficult to compute the dimension of the algebraic set that it defines.
947:
134:
1925:
1422:
777:
759:
157:
855:
1970:
1915:
1460:
511:
507:
494:
224:
215:
2052:
2037:
1230:
1160:
842:
1170:
All the definitions of the previous section apply, with the change that, when
2173:
2042:
1535:
858:
computation to compute the dimension of the algebraic set defined by a given
720:
684:
145:
367:
is replaced by another ideal having the same zeros (that is having the same
144:. Some are restricted to algebraic varieties while others apply also to any
2062:
2027:
1920:
1763:
755:
498:
153:
16:
Measure of a mathematical object studied in the field of algebraic geometry
1855:
Probabilistic
Algorithm for Computing the Dimension of Real Algebraic Sets
1841:, Algorithms and Computation in Mathematics, vol. 10, Springer-Verlag
2147:
1930:
1693:
586:
522:
This is the transcription of the preceding definition in the language of
122:
2142:
2022:
807:
This is the algebraic translation of the fact that the intersection of
751:
678:
This rephrases the previous definition into a more geometric language.
601:
2123:
2032:
1945:
1896:
1430:
149:
670:
is a variety, the Krull dimension of the local ring at any point of
24:
2047:
2010:
1935:
737:
657:
is irreducible, it turns out that all the local rings at points of
1399:{\displaystyle x_{1}^{\min(e_{1},1)}\cdots x_{n}^{\min(e_{n},1)}.}
1037:
This allows to prove easily that the dimension is invariant under
2057:
1832:
Basu, Saugata; Pollack, Richard; Roy, Marie-Françoise (2003),
1059:
defined as the set of the common zeros of a homogeneous ideal
578:{\displaystyle p_{0}\subset p_{1}\subset \ldots \subset p_{d}}
476:{\displaystyle V_{0}\subset V_{1}\subset \ldots \subset V_{d}}
2014:
1467:, the real dimension is one of the following equal integers:
798:
and the maximal length of the regular sequences contained in
526:, the Krull dimension being the maximal length of the chains
493:
This definition generalizes a property of the dimension of a
1865:
789:
This the algebraic translation of the preceding definition.
736:
This is the algebraic analogue to the fact that a connected
1044:
355:
is replaced by another algebraically closed extension of
343:
is any of the following integers. It does not change if
817:
general hypersurfaces is an algebraic set of dimension
163:
766:
which is reduced to a nonzero finite number of points.
1706:
1670:
1650:
1626:
1606:
1583:
1544:
1520:
1500:
1477:
1315:
1302:{\displaystyle {x_{1}}^{e_{1}}\cdots {x_{n}}^{e_{n}}}
1243:
1212:
1071:
980:
960:
932:
904:
876:
700:
This relates the dimension of a variety to that of a
643:
621:
532:
430:
410:
296:
232:
1406:Then the dimension is the maximal size of a subset
1309:is replaced by the product of the variables in it:
974:is the set of all leading monomials of elements of
484:of distinct nonempty (irreducible) subvarieties of
49:. Unsourced material may be challenged and removed.
1751:
1676:
1656:
1632:
1612:
1589:
1569:
1526:
1506:
1483:
1398:
1301:
1218:
1131:
986:
966:
938:
910:
890:
649:
627:
577:
475:
416:
316:
282:
2171:
1366:
1326:
160:, while other are related to such an embedding.
1197:
1831:
762:which are needed to have an intersection with
613:This definition shows that the dimension is a
1881:
1852:Ivan, Bannwarth; Mohab, Safey El Din (2015),
661:have the same Krull dimension (see ); thus:
148:. Some are intrinsic, as independent of any
137:may be defined in various equivalent ways.
1888:
1874:
1851:
1417:This algorithm is implemented in several
1233:of the ideal generated by the equations.
109:Learn how and when to remove this message
1491:is the dimension of its Zariski closure.
1188:is one less than the Krull dimension of
1684:-dimensional subspace with a non-empty
1045:Dimension of a projective algebraic set
2172:
1692:For an algebraic set defined over the
1869:
1835:Algorithms in Real Algebraic Geometry
1455:of a set of real points, typically a
1003:defined by this Stanley–Reisner ring.
841:The degree of the denominator of the
164:Dimension of an affine algebraic set
47:adding citations to reliable sources
18:
1752:{\displaystyle x^{2}+y^{2}+z^{2}=0}
1206:over an algebraically closed field
719:is a variety, the dimension of the
600:The maximal Krull dimension of the
58:"Dimension of an algebraic variety"
13:
14:
2201:
1440:
331:of the polynomial functions over
1954:
23:
34:needs additional citations for
1845:
1825:
1815:
1798:
1558:
1545:
1388:
1369:
1348:
1329:
1204:system of polynomial equations
1126:
1081:
1016:is an algebraic variety), the
860:system of polynomial equations
274:
242:
190:algebraically closed extension
1:
1895:
1791:
683:The maximal dimension of the
1198:Computation of the dimension
7:
1769:
867:The Krull dimension of the
10:
2206:
1781:Dimension theory (algebra)
1463:. For a semialgebraic set
1459:, is the dimension of its
1444:
2156:
2135:
2071:
2009:
1963:
1952:
1903:
204:is the set of the common
1776:Dimension (vector space)
1419:computer algebra systems
1163:of the polynomials over
1057:projective algebraic set
776:The maximal length of a
1620:is the maximal integer
1514:is the maximal integer
1433:, this is the function
1425:, this is the function
1012:is a prime ideal (i.e.
854:This allows, through a
794:The difference between
780:in the coordinate ring
702:differentiable manifold
152:of the variety into an
1753:
1678:
1658:
1634:
1614:
1600:The real dimension of
1591:
1571:
1528:
1508:
1494:The real dimension of
1485:
1471:The real dimension of
1400:
1303:
1220:
1133:
1039:birational equivalence
988:
968:
940:
912:
892:
651:
629:
579:
477:
418:
318:
284:
214:of the elements of an
1786:Dimension of a scheme
1754:
1679:
1659:
1640:such that there is a
1635:
1615:
1592:
1572:
1534:such that there is a
1529:
1509:
1486:
1401:
1304:
1221:
1134:
1065:in a polynomial ring
999:The dimension of the
989:
969:
941:
913:
893:
743:Zariski tangent space
704:. More precisely, if
685:tangent vector spaces
652:
630:
580:
478:
419:
319:
317:{\displaystyle A=R/I}
285:
2072:Dimensions by number
1704:
1668:
1648:
1624:
1604:
1581:
1570:{\displaystyle ^{d}}
1542:
1518:
1498:
1475:
1313:
1241:
1210:
1069:
1018:transcendence degree
978:
958:
930:
902:
874:
869:Stanley–Reisner ring
721:tangent vector space
641:
619:
530:
428:
408:
384:linearly independent
294:
283:{\displaystyle R=K.}
230:
197:affine algebraic set
125:and specifically in
43:improve this article
2180:Algebraic varieties
1392:
1352:
1132:{\displaystyle R=K}
946:for any admissible
891:{\displaystyle R/J}
524:commutative algebra
387:linear combinations
337:. The dimension of
142:commutative algebra
2001:Degrees of freedom
1904:Dimensional spaces
1749:
1674:
1654:
1630:
1610:
1587:
1567:
1524:
1504:
1481:
1396:
1356:
1316:
1299:
1216:
1129:
1026:field of fractions
1001:simplicial complex
984:
964:
936:
908:
888:
830:Hilbert polynomial
828:The degree of the
647:
625:
615:local property if
575:
473:
414:
403:The maximal length
314:
280:
127:algebraic geometry
2167:
2166:
1976:Lebesgue covering
1941:Algebraic variety
1810:978-0-201-40751-8
1698:algebraic surface
1677:{\displaystyle d}
1657:{\displaystyle S}
1633:{\displaystyle d}
1613:{\displaystyle S}
1590:{\displaystyle S}
1527:{\displaystyle d}
1507:{\displaystyle S}
1484:{\displaystyle S}
1457:semialgebraic set
1447:Complex dimension
1421:. For example in
1219:{\displaystyle K}
987:{\displaystyle I}
967:{\displaystyle I}
948:monomial ordering
939:{\displaystyle I}
911:{\displaystyle J}
650:{\displaystyle V}
628:{\displaystyle V}
604:at the points of
417:{\displaystyle d}
392:The dimension of
135:algebraic variety
119:
118:
111:
93:
2197:
2190:Computer algebra
1964:Other dimensions
1958:
1926:Projective space
1890:
1883:
1876:
1867:
1866:
1860:
1859:
1849:
1843:
1842:
1840:
1829:
1823:
1819:
1813:
1802:
1758:
1756:
1755:
1750:
1742:
1741:
1729:
1728:
1716:
1715:
1683:
1681:
1680:
1675:
1663:
1661:
1660:
1655:
1639:
1637:
1636:
1631:
1619:
1617:
1616:
1611:
1596:
1594:
1593:
1588:
1576:
1574:
1573:
1568:
1566:
1565:
1533:
1531:
1530:
1525:
1513:
1511:
1510:
1505:
1490:
1488:
1487:
1482:
1466:
1405:
1403:
1402:
1397:
1391:
1381:
1380:
1364:
1351:
1341:
1340:
1324:
1308:
1306:
1305:
1300:
1298:
1297:
1296:
1295:
1285:
1284:
1283:
1269:
1268:
1267:
1266:
1256:
1255:
1254:
1225:
1223:
1222:
1217:
1193:
1187:
1181:
1175:
1158:
1144:
1138:
1136:
1135:
1130:
1125:
1124:
1106:
1105:
1093:
1092:
1064:
1054:
1031:
1023:
1015:
1011:
993:
991:
990:
985:
973:
971:
970:
965:
945:
943:
942:
937:
917:
915:
914:
909:
897:
895:
894:
889:
884:
848:
835:
822:
816:
801:
797:
783:
778:regular sequence
765:
760:general position
730:
718:
709:
694:
673:
669:
660:
656:
654:
653:
648:
634:
632:
631:
626:
607:
594:
584:
582:
581:
576:
574:
573:
555:
554:
542:
541:
516:
487:
482:
480:
479:
474:
472:
471:
453:
452:
440:
439:
423:
421:
420:
415:
397:
382:are replaced by
381:
366:
360:
354:
349:is enlarged, if
348:
342:
336:
323:
321:
320:
315:
310:
289:
287:
286:
281:
273:
272:
254:
253:
222:
213:
203:
187:
173:
158:projective space
114:
107:
103:
100:
94:
92:
51:
27:
19:
2205:
2204:
2200:
2199:
2198:
2196:
2195:
2194:
2170:
2169:
2168:
2163:
2152:
2131:
2067:
2005:
1959:
1950:
1916:Euclidean space
1899:
1894:
1864:
1863:
1850:
1846:
1838:
1830:
1826:
1820:
1816:
1803:
1799:
1794:
1772:
1737:
1733:
1724:
1720:
1711:
1707:
1705:
1702:
1701:
1669:
1666:
1665:
1649:
1646:
1645:
1625:
1622:
1621:
1605:
1602:
1601:
1582:
1579:
1578:
1561:
1557:
1543:
1540:
1539:
1519:
1516:
1515:
1499:
1496:
1495:
1476:
1473:
1472:
1464:
1461:Zariski closure
1449:
1443:
1376:
1372:
1365:
1360:
1336:
1332:
1325:
1320:
1314:
1311:
1310:
1291:
1287:
1286:
1279:
1275:
1274:
1273:
1262:
1258:
1257:
1250:
1246:
1245:
1244:
1242:
1239:
1238:
1211:
1208:
1207:
1200:
1189:
1183:
1177:
1171:
1146:
1140:
1120:
1116:
1101:
1097:
1088:
1084:
1070:
1067:
1066:
1060:
1050:
1047:
1029:
1021:
1013:
1009:
979:
976:
975:
959:
956:
955:
931:
928:
927:
903:
900:
899:
880:
875:
872:
871:
846:
833:
818:
808:
799:
795:
781:
763:
728:
716:
705:
692:
689:singular points
671:
667:
658:
642:
639:
638:
635:is irreducible.
620:
617:
616:
605:
590:
569:
565:
550:
546:
537:
533:
531:
528:
527:
514:
512:coordinate ring
508:Krull dimension
495:Euclidean space
485:
467:
463:
448:
444:
435:
431:
429:
426:
425:
409:
406:
405:
393:
380:
372:
362:
356:
350:
344:
338:
332:
306:
295:
292:
291:
268:
264:
249:
245:
231:
228:
227:
225:polynomial ring
218:
209:
199:
179:
169:
166:
115:
104:
98:
95:
52:
50:
40:
28:
17:
12:
11:
5:
2203:
2193:
2192:
2187:
2182:
2165:
2164:
2157:
2154:
2153:
2151:
2150:
2145:
2139:
2137:
2133:
2132:
2130:
2129:
2121:
2116:
2111:
2106:
2101:
2096:
2091:
2086:
2081:
2075:
2073:
2069:
2068:
2066:
2065:
2060:
2055:
2053:Cross-polytope
2050:
2045:
2040:
2038:Hyperrectangle
2035:
2030:
2025:
2019:
2017:
2007:
2006:
2004:
2003:
1998:
1993:
1988:
1983:
1978:
1973:
1967:
1965:
1961:
1960:
1953:
1951:
1949:
1948:
1943:
1938:
1933:
1928:
1923:
1918:
1913:
1907:
1905:
1901:
1900:
1893:
1892:
1885:
1878:
1870:
1862:
1861:
1844:
1824:
1814:
1796:
1795:
1793:
1790:
1789:
1788:
1783:
1778:
1771:
1768:
1748:
1745:
1740:
1736:
1732:
1727:
1723:
1719:
1714:
1710:
1690:
1689:
1673:
1653:
1629:
1609:
1598:
1586:
1564:
1560:
1556:
1553:
1550:
1547:
1523:
1503:
1492:
1480:
1453:real dimension
1442:
1441:Real dimension
1439:
1395:
1390:
1387:
1384:
1379:
1375:
1371:
1368:
1363:
1359:
1355:
1350:
1347:
1344:
1339:
1335:
1331:
1328:
1323:
1319:
1294:
1290:
1282:
1278:
1272:
1265:
1261:
1253:
1249:
1231:Hilbert series
1215:
1199:
1196:
1161:graded algebra
1128:
1123:
1119:
1115:
1112:
1109:
1104:
1100:
1096:
1091:
1087:
1083:
1080:
1077:
1074:
1046:
1043:
1035:
1034:
1005:
996:
983:
963:
935:
907:
887:
883:
879:
852:
851:
843:Hilbert series
838:
805:
804:
787:
786:
769:
768:
750:The number of
734:
733:
725:singular point
698:
697:
676:
675:
646:
624:
611:
610:
572:
568:
564:
561:
558:
553:
549:
545:
540:
536:
520:
519:
491:
490:
470:
466:
462:
459:
456:
451:
447:
443:
438:
434:
424:of the chains
413:
376:
313:
309:
305:
302:
299:
279:
276:
271:
267:
263:
260:
257:
252:
248:
244:
241:
238:
235:
165:
162:
117:
116:
31:
29:
22:
15:
9:
6:
4:
3:
2:
2202:
2191:
2188:
2186:
2183:
2181:
2178:
2177:
2175:
2162:
2161:
2155:
2149:
2146:
2144:
2141:
2140:
2138:
2134:
2128:
2126:
2122:
2120:
2117:
2115:
2112:
2110:
2107:
2105:
2102:
2100:
2097:
2095:
2092:
2090:
2087:
2085:
2082:
2080:
2077:
2076:
2074:
2070:
2064:
2061:
2059:
2056:
2054:
2051:
2049:
2046:
2044:
2043:Demihypercube
2041:
2039:
2036:
2034:
2031:
2029:
2026:
2024:
2021:
2020:
2018:
2016:
2012:
2008:
2002:
1999:
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60: –
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54:Find sources:
48:
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38:
37:
32:This article
30:
26:
21:
20:
2158:
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2063:Hyperpyramid
2028:Hypersurface
1940:
1921:Affine space
1911:Vector space
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1764:hypersurface
1761:
1700:of equation
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587:prime ideals
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499:vector space
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41:Please help
36:verification
33:
2148:Codimension
2127:-dimensions
2048:Hypersphere
1931:Free module
752:hyperplanes
723:at any non
687:at the non
602:local rings
123:mathematics
2174:Categories
2143:Hyperspace
2023:Hyperplane
1792:References
1642:projection
1445:See also:
1145:, and let
389:of them.
99:April 2016
69:newspapers
2185:Dimension
2033:Hypercube
2011:Polytopes
1991:Minkowski
1986:Hausdorff
1981:Inductive
1946:Spacetime
1897:Dimension
1431:Macaulay2
1427:Groebner,
1354:⋯
1271:⋯
1111:…
563:⊂
560:…
557:⊂
544:⊂
461:⊂
458:…
455:⊂
442:⊂
259:…
150:embedding
131:dimension
2160:Category
2136:See also
1936:Manifold
1770:See also
1686:interior
1202:Given a
738:manifold
329:-algebra
2058:Simplex
1996:Fractal
1664:over a
1429:and in
1159:be the
1024:of the
922:of the
920:radical
918:is the
510:of the
369:radical
361:and if
324:be the
83:scholar
2015:shapes
1808:
898:where
188:be an
178:, and
154:affine
133:of an
129:, the
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2119:Eight
2114:Seven
2094:Three
1971:Krull
1839:(PDF)
1822:2015.
1694:reals
1423:Maple
1055:be a
1020:over
950:(the
497:or a
223:in a
216:ideal
206:zeros
176:field
174:be a
90:JSTOR
76:books
2104:Five
2099:Four
2079:Zero
2013:and
1806:ISBN
1451:The
1049:Let
506:The
290:Let
168:Let
62:news
2109:Six
2089:Two
2084:One
1644:of
1577:in
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1435:dim
1367:min
1327:min
1176:or
1028:of
1008:If
954:of
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832:of
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758:in
754:or
745:).
727:of
715:If
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