25:
1089:
698:
1218:
1565:
1325:
893:
2644:
442:
Note, that the
Bernstein–Sato polynomial can be computed algorithmically. However, such computations are hard in general. There are implementations of related algorithms in computer algebra systems RISA/Asir,
904:
786:
1780:
2096:
548:
365:
2200:
for such cases has so far proved prohibitively cumbersome. Devising ways to bypass the combinatorial explosion of the brute force algorithm would be of great value in such applications.
1424:
2102:
having 2-6 scalar components, and the pair of polynomials having orders 2 and 3. Unfortunately, a brute force determination of the corresponding differential operators
1946:
2198:
2149:
1893:
559:
1652:
1616:
401:
271:
242:
213:
1106:
2794:
1435:
1243:
2003:)). However, the most interesting cases require a simple generalization of the Bernstein-Sato functional equation to the product of two polynomials
458:) presented algorithms to compute the Bernstein–Sato polynomial of an affine variety together with an implementation in the computer algebra system
2247:
Andres, Daniel; Levandovskyy, Viktor; Martín-Morales, Jorge (2009). "Principal intersection and bernstein-sato polynomial of an affine variety".
46:
797:
1084:{\displaystyle \prod _{j=1}^{r}\partial _{x_{j}}^{n_{j}}\quad f(x)^{s+1}=\prod _{j=1}^{r}\prod _{i=1}^{n_{j}}(n_{j}s+i)\quad f(x)^{s}}
2344:(1971). "Modules over a ring of differential operators. Study of the fundamental solutions of equations with constant coefficients".
1987:
The
Bernstein-Sato functional equation is used in computations of some of the more complex kinds of singular integrals occurring in
715:
2590:
2505:
1685:
2006:
2252:
1833:) is non-negative the inverse can be constructed using the Bernstein–Sato polynomial by taking the constant term of the
481:
1995:). Such computations are needed for precision measurements in elementary particle physics as practiced for instance at
1970:
this follows from the fact that every polynomial has a distributional inverse, which is proved in the paragraph above.
2481:
2458:
2323:
2278:
74:
1951:
279:
2855:
2792:
Tkachov, Fyodor V. (1997). "Algebraic algorithms for multiloop calculations. The first 15 years. What's next?".
51:
2296:
Berkesch, Christine; Leykin, Anton (2010). "Algorithms for
Bernstein--Sato polynomials and multiplier ideals".
2582:
1364:
1898:
2473:
1981:
1666:
693:{\displaystyle \sum _{i=1}^{n}\partial _{i}^{2}f(x)^{s+1}=4(s+1)\left(s+{\frac {n}{2}}\right)f(x)^{s}}
1572:
2154:
2105:
2529:
1860:
420:
The
Bernstein–Sato polynomial can also be defined for products of powers of several polynomials (
33:
2614:
2387:
2755:
1621:
2714:
2712:; Shintani, Takuro (1974). "On zeta functions associated with prehomogeneous vector spaces".
1959:
1955:
1659:
96:
2850:
2813:
2781:
2743:
2683:
2653:
2627:
2600:
2558:
2538:
2515:
2426:
2406:
2365:
1988:
1213:{\displaystyle b(s)=\prod _{j=1}^{r}\prod _{i=1}^{n_{j}}\left(s+{\frac {i}{n_{j}}}\right).}
152:
140:
136:
1592:
377:
247:
218:
189:
8:
2382:
1963:
1674:
469:) described some of the algorithms for computing Bernstein–Sato polynomials by computer.
428:
2817:
2657:
2542:
2410:
2829:
2803:
2731:
2671:
2562:
2430:
2396:
2369:
2329:
2301:
2284:
2256:
144:
2825:
2692:
2639:
2697:
2586:
2501:
2477:
2454:
2373:
2319:
2298:
Proceedings of the 2010 International
Symposium on Symbolic and Algebraic Computation
2274:
2249:
Proceedings of the 2009 international symposium on
Symbolic and algebraic computation
1967:
1834:
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2434:
2414:
2353:
2341:
2311:
2288:
2266:
1576:
454:
Daniel Andres, Viktor
Levandovskyy, and Jorge Martín-Morales (
371:
171:
148:
100:
56:
2527:(1976). "B-functions and holonomic systems. Rationality of roots of B-functions".
2500:. Vol. 1. Providence, R.I.: American Mathematical Society. pp. 597–607.
2777:
2739:
2679:
2623:
2596:
2554:
2511:
2422:
2361:
414:
1560:{\displaystyle \Omega (\det(t_{ij})^{s})=s(s+1)\cdots (s+n-1)\det(t_{ij})^{s-1}}
1320:{\displaystyle (s+1)\left(s+{\frac {5}{6}}\right)\left(s+{\frac {7}{6}}\right).}
2645:
Proceedings of the
National Academy of Sciences of the United States of America
2450:
2385:; Saito, Morihiko (2006). "Bernstein-Sato polynomials of arbitrary varieties".
432:
39:
2772:
2418:
424:). In this case it is a product of linear factors with rational coefficients.
2844:
2491:
1973:
2581:. Translations of Mathematical Monographs. Vol. 217. Providence, R.I.:
2315:
2270:
2701:
2666:
2442:
163:
2472:. London Mathematical Society Student Texts. Vol. 33. Cambridge, UK:
215:
is a polynomial in several variables, then there is a non-zero polynomial
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88:
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2751:
2735:
2709:
2635:
2550:
2357:
108:
2675:
2401:
444:
2727:
2495:
439:) generalized the Bernstein–Sato polynomial to arbitrary varieties.
413:
proved that all roots of the
Bernstein–Sato polynomial are negative
888:{\displaystyle f(x)=x_{1}^{n_{1}}x_{2}^{n_{2}}\cdots x_{r}^{n_{r}}}
459:
448:
404:
2306:
2261:
1817:) is a polynomial, not identically zero, then it has an inverse
2640:"On zeta functions associated with prehomogeneous vector spaces"
2246:
455:
2786:
the
English translation of Sato's lecture from Shintani's note
2610:"Proximité évanescente. I. La structure polaire d'un D-module"
403:. Its existence can be shown using the notion of holonomic
1996:
2221:
has zeros then there are distributions whose product with
2217:
Warning: The inverse is not unique in general, because if
2756:"Theory of prehomogeneous vector spaces (algebraic part)"
2449:. Perspectives in Mathematics. Vol. 2. Boston, MA:
781:{\displaystyle b(s)=(s+1)\left(s+{\frac {n}{2}}\right).}
2497:
Quantum fields and strings: A course for mathematicians
1980:) showed how to use the Bernstein polynomial to define
1775:{\displaystyle f(x)^{s}={1 \over b(s)}P(s)f(x)^{s+1}.}
465:
Christine Berkesch and Anton Leykin (
2157:
2108:
2009:
1901:
1863:
1688:
1624:
1595:
1438:
1367:
1346:
variables, then the Bernstein–Sato polynomial of det(
1246:
1109:
907:
800:
718:
562:
484:
380:
282:
250:
221:
192:
2091:{\displaystyle (f_{1}(x))^{s_{1}}(f_{2}(x))^{s_{2}}}
2225:is zero, and adding one of these to an inverse of
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2143:
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1887:
1774:
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1610:
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1319:
1212:
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780:
692:
543:{\displaystyle f(x)=x_{1}^{2}+\cdots +x_{n}^{2}\,}
542:
395:
359:
265:
236:
207:
2380:
436:
2842:
1522:
1445:
2494:(1999). "Note on dimensional regularization".
2295:
466:
2708:
2634:
374:of smallest degree amongst such polynomials
360:{\displaystyle P(s)f(x)^{s+1}=b(s)f(x)^{s}.}
181:
116:
112:
16:Polynomial related to differential operators
1984:rigorously, in the massive Euclidean case.
162:) gives an elementary introduction, while
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2771:
2691:
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2523:
2400:
2340:
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2260:
539:
410:
175:
104:
75:Learn how and when to remove this message
2467:
2346:Functional Analysis and Its Applications
1821:that is a distribution; in other words,
159:
2791:
2490:
2000:
1992:
1977:
273:with polynomial coefficients such that
2843:
2607:
1419:{\displaystyle (s+1)(s+2)\cdots (s+n)}
421:
2441:
1849: = −1. For arbitrary
1800:) is zero for a non-negative integer
370:The Bernstein–Sato polynomial is the
167:
55:. Parenthetical referencing has been
2750:
1825: = 1 as distributions. If
1658:with non-negative real part, can be
120:
18:
2253:Association for Computing Machinery
706:so the Bernstein–Sato polynomial is
111: and Takuro Shintani (
13:
1941:{\displaystyle {\bar {f}}(x)f(x).}
1618:is a non-negative polynomial then
1439:
930:
585:
135:, though it is not related to the
14:
2867:
2579:D-modules and microlocal calculus
1575:, which in turn follows from the
1226:The Bernstein–Sato polynomial of
59:; convert to shortened footnotes.
23:
2470:A primer of algebraic D-modules
1583:
1061:
958:
178:) give more advanced accounts.
99:, introduced independently by
2468:Coutinho, Severino C. (1995).
2211:
2193:{\displaystyle b(s_{1},s_{2})}
2187:
2161:
2144:{\displaystyle P(s_{1},s_{2})}
2138:
2112:
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1:
2826:10.1016/S0168-9002(97)00110-1
2583:American Mathematical Society
2240:
1888:{\displaystyle {\bar {f}}(x)}
1952:Malgrange–Ehrenpreis theorem
244:and a differential operator
158:Severino Coutinho (
7:
2760:Nagoya Mathematical Journal
2638:; Shintani, Takuro (1972).
1788:It may have poles whenever
472:
95:is a polynomial related to
10:
2872:
2520:(Princeton, NJ, 1996/1997)
2474:Cambridge University Press
1982:dimensional regularization
123:. It is also known as the
2773:10.1017/s0027763000003214
2419:10.1112/S0010437X06002193
1991:Fyodor Tkachov (
182:Definition and properties
143:. It has applications to
93:Bernstein–Sato polynomial
2795:Nucl. Instrum. Methods A
2530:Inventiones Mathematicae
2204:
1999:(see the papers citing (
1673:by repeatedly using the
1654:, initially defined for
1647:{\displaystyle f(x)^{s}}
2608:Sabbah, Claude (1987).
2316:10.1145/1837934.1837958
2271:10.1145/1576702.1576735
2856:Differential operators
2667:10.1073/pnas.69.5.1081
2615:Compositio Mathematica
2388:Compositio Mathematica
2229:is another inverse of
2194:
2145:
2092:
1942:
1889:
1776:
1660:analytically continued
1648:
1612:
1573:Cayley's omega process
1561:
1420:
1321:
1214:
1173:
1145:
1085:
1035:
1007:
928:
889:
782:
694:
583:
544:
397:
361:
267:
238:
209:
97:differential operators
32:This article includes
2715:Annals of Mathematics
2195:
2146:
2093:
1960:constant coefficients
1956:differential operator
1943:
1895:times the inverse of
1890:
1777:
1649:
1613:
1562:
1421:
1322:
1215:
1146:
1125:
1086:
1008:
987:
908:
890:
783:
695:
563:
545:
398:
362:
268:
239:
210:
137:Bernstein polynomials
2255:. pp. 231–238.
2155:
2106:
2007:
1989:quantum field theory
1899:
1861:
1686:
1669:-valued function of
1622:
1611:{\displaystyle f(x)}
1593:
1436:
1365:
1244:
1107:
905:
798:
716:
560:
482:
427:Nero Budur,
396:{\displaystyle b(s)}
378:
280:
266:{\displaystyle P(s)}
248:
237:{\displaystyle b(s)}
219:
208:{\displaystyle f(x)}
190:
172:Masaki Kashiwara
153:quantum field theory
141:approximation theory
133:Bernstein polynomial
101:Joseph Bernstein
2818:1997NIMPA.389..309T
2658:1972PNAS...69.1081S
2543:1976InMat..38...33K
2447:Algebraic D-Modules
2411:2004math......8408B
2300:. pp. 99–106.
1675:functional equation
1430:which follows from
957:
884:
859:
837:
598:
538:
514:
2551:10.1007/BF01390168
2358:10.1007/BF01076413
2190:
2141:
2088:
1968:Fourier transforms
1954:states that every
1938:
1885:
1772:
1644:
1608:
1557:
1416:
1317:
1210:
1081:
929:
885:
863:
838:
816:
778:
690:
584:
540:
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500:
393:
357:
263:
234:
205:
145:singularity theory
40:properly formatted
2718:. Second Series.
2592:978-0-8218-2766-6
2575:Kashiwara, Masaki
2525:Kashiwara, Masaki
2507:978-0-8218-2012-4
2342:Bernstein, Joseph
1974:Pavel Etingof
1911:
1873:
1835:Laurent expansion
1730:
1307:
1281:
1200:
768:
664:
85:
84:
77:
2863:
2837:
2811:
2802:(1–2): 309–313.
2788:
2775:
2747:
2705:
2695:
2669:
2652:(5): 1081–1082.
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2604:
2570:
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2048:
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2045:
2022:
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1964:Green's function
1947:
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1653:
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1617:
1615:
1614:
1609:
1577:Capelli identity
1571:where Ω is
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415:rational numbers
411:Kashiwara (1976)
402:
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372:monic polynomial
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164:Armand Borel
149:monodromy theory
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49:this article by
34:inline citations
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2722:(1): 131–170.
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2622:(3): 283–328.
2605:
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2492:Etingof, Pavel
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2451:Academic Press
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2395:(3): 779–797.
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1458:
1454:
1450:
1447:
1444:
1441:
1428:
1427:
1426:
1415:
1412:
1409:
1406:
1403:
1400:
1397:
1394:
1391:
1388:
1385:
1382:
1379:
1376:
1373:
1370:
1357:
1356:
1350:
1337:
1330:
1329:
1328:
1327:
1316:
1312:
1306:
1303:
1298:
1295:
1291:
1286:
1280:
1277:
1272:
1269:
1265:
1261:
1258:
1255:
1252:
1249:
1236:
1235:
1223:
1222:
1221:
1220:
1209:
1205:
1197:
1193:
1189:
1184:
1181:
1177:
1169:
1165:
1159:
1156:
1153:
1149:
1143:
1138:
1135:
1132:
1128:
1124:
1121:
1118:
1115:
1112:
1099:
1098:
1094:
1093:
1092:
1091:
1078:
1074:
1070:
1067:
1064:
1060:
1057:
1054:
1051:
1046:
1042:
1038:
1031:
1027:
1021:
1018:
1015:
1011:
1005:
1000:
997:
994:
990:
986:
981:
978:
975:
971:
967:
964:
961:
953:
949:
941:
937:
932:
926:
921:
918:
915:
911:
897:
896:
880:
876:
870:
866:
862:
855:
851:
845:
841:
833:
829:
823:
819:
815:
812:
809:
806:
803:
791:
790:
789:
788:
777:
773:
767:
764:
759:
756:
752:
748:
745:
742:
739:
736:
733:
730:
727:
724:
721:
708:
707:
703:
702:
701:
700:
687:
683:
679:
676:
673:
669:
663:
660:
655:
652:
648:
644:
641:
638:
635:
632:
629:
626:
621:
618:
615:
611:
607:
604:
601:
596:
591:
587:
581:
576:
573:
570:
566:
552:
551:
536:
531:
527:
523:
520:
517:
512:
507:
503:
499:
496:
493:
490:
487:
474:
471:
433:Morihiko Saito
431:, and
429:Mircea Mustață
392:
389:
386:
383:
368:
367:
356:
351:
347:
343:
340:
337:
334:
331:
328:
325:
322:
317:
314:
311:
307:
303:
300:
297:
294:
291:
288:
285:
262:
259:
256:
253:
233:
230:
227:
224:
204:
201:
198:
195:
183:
180:
109:Mikio Sato
83:
82:
31:
29:
22:
15:
9:
6:
4:
3:
2:
2868:
2857:
2854:
2852:
2849:
2848:
2846:
2835:
2831:
2827:
2823:
2819:
2815:
2810:
2805:
2801:
2797:
2796:
2790:
2787:
2783:
2779:
2774:
2769:
2765:
2761:
2757:
2753:
2749:
2745:
2741:
2737:
2733:
2729:
2725:
2721:
2717:
2716:
2711:
2707:
2703:
2699:
2694:
2689:
2685:
2681:
2677:
2673:
2668:
2663:
2659:
2655:
2651:
2647:
2646:
2641:
2637:
2633:
2629:
2625:
2621:
2617:
2616:
2611:
2606:
2602:
2598:
2594:
2588:
2584:
2580:
2576:
2572:
2568:
2564:
2560:
2556:
2552:
2548:
2544:
2540:
2536:
2532:
2531:
2526:
2522:
2517:
2513:
2509:
2503:
2499:
2498:
2493:
2489:
2485:
2483:0-521-55908-1
2479:
2475:
2471:
2466:
2462:
2460:0-12-117740-8
2456:
2452:
2448:
2444:
2443:Borel, Armand
2440:
2436:
2432:
2428:
2424:
2420:
2416:
2412:
2408:
2403:
2398:
2394:
2390:
2389:
2384:
2381:Budur, Nero;
2379:
2375:
2371:
2367:
2363:
2359:
2355:
2352:(2): 89–101.
2351:
2347:
2343:
2339:
2335:
2331:
2327:
2325:9781450301503
2321:
2317:
2313:
2308:
2303:
2299:
2294:
2290:
2286:
2282:
2280:9781605586090
2276:
2272:
2268:
2263:
2258:
2254:
2250:
2245:
2244:
2232:
2228:
2224:
2220:
2214:
2210:
2182:
2178:
2174:
2169:
2165:
2158:
2133:
2129:
2125:
2120:
2116:
2109:
2101:
2081:
2077:
2065:
2057:
2053:
2042:
2038:
2026:
2018:
2014:
2002:
1998:
1994:
1990:
1986:
1983:
1979:
1975:
1972:
1969:
1965:
1961:
1957:
1953:
1949:
1935:
1929:
1923:
1917:
1905:
1879:
1867:
1856:
1852:
1848:
1844:
1840:
1836:
1832:
1828:
1824:
1820:
1816:
1812:
1808:
1807:
1803:
1799:
1796: +
1795:
1791:
1787:
1786:
1769:
1764:
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1758:
1750:
1744:
1738:
1732:
1723:
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1713:
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1689:
1682:
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1570:
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1536:
1533:
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1469:
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1432:
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1404:
1398:
1392:
1389:
1386:
1377:
1374:
1371:
1361:
1360:
1359:
1358:
1355:) is given by
1353:
1349:
1345:
1340:
1336:
1332:
1331:
1314:
1310:
1304:
1301:
1296:
1293:
1289:
1284:
1278:
1275:
1270:
1267:
1263:
1256:
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1250:
1240:
1239:
1238:
1237:
1233:
1230: +
1229:
1225:
1224:
1207:
1203:
1195:
1191:
1187:
1182:
1179:
1175:
1167:
1163:
1157:
1154:
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1147:
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1130:
1126:
1122:
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1110:
1103:
1102:
1101:
1100:
1096:
1095:
1076:
1068:
1062:
1055:
1052:
1049:
1044:
1040:
1029:
1025:
1019:
1016:
1013:
1009:
1003:
998:
995:
992:
988:
984:
979:
976:
973:
965:
959:
951:
947:
939:
935:
924:
919:
916:
913:
909:
901:
900:
899:
898:
878:
874:
868:
864:
860:
853:
849:
843:
839:
831:
827:
821:
817:
813:
807:
801:
793:
792:
775:
771:
765:
762:
757:
754:
750:
743:
740:
737:
731:
725:
719:
712:
711:
710:
709:
705:
704:
685:
677:
671:
667:
661:
658:
653:
650:
646:
639:
636:
633:
627:
624:
619:
616:
613:
605:
599:
594:
589:
579:
574:
571:
568:
564:
556:
555:
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521:
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515:
510:
505:
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497:
491:
485:
477:
476:
470:
468:
463:
461:
457:
452:
450:
446:
440:
438:
434:
430:
425:
423:
418:
416:
412:
408:
406:
387:
381:
373:
354:
349:
341:
335:
329:
323:
320:
315:
312:
309:
301:
295:
289:
283:
276:
275:
274:
257:
251:
228:
222:
199:
193:
179:
177:
173:
169:
165:
161:
156:
154:
150:
146:
142:
138:
134:
130:
126:
122:
118:
114:
110:
106:
102:
98:
94:
90:
79:
76:
68:
58:
53:
48:
43:
41:
38:they are not
35:
30:
21:
20:
2799:
2793:
2785:
2763:
2759:
2719:
2713:
2649:
2643:
2619:
2613:
2578:
2537:(1): 33–53.
2534:
2528:
2496:
2469:
2446:
2402:math/0408408
2392:
2386:
2349:
2345:
2297:
2248:
2230:
2226:
2222:
2218:
2213:
2099:
2001:Tkachov 1997
1966:. By taking
1857:) just take
1854:
1850:
1846:
1842:
1838:
1830:
1826:
1822:
1818:
1814:
1810:
1801:
1797:
1793:
1789:
1670:
1667:distribution
1655:
1584:Applications
1351:
1347:
1343:
1338:
1334:
1231:
1227:
464:
453:
441:
426:
419:
409:
369:
185:
157:
132:
129:b-polynomial
128:
124:
92:
86:
71:
62:
37:
2851:Polynomials
2752:Sato, Mikio
2710:Sato, Mikio
2636:Sato, Mikio
1664:meromorphic
422:Sabbah 1987
121:Sato (1990)
89:mathematics
2845:Categories
2241:References
131:, and the
125:b-function
57:deprecated
2754:(1990) .
2374:124605141
2307:1002.1475
2262:1002.3644
1909:¯
1871:¯
1550:−
1514:−
1499:⋯
1440:Ω
1399:⋯
1148:∏
1127:∏
1010:∏
989:∏
931:∂
910:∏
861:⋯
586:∂
565:∑
519:⋯
445:Macaulay2
405:D-modules
2834:37109930
2766:: 1–34.
2702:16591979
2577:(2003).
2567:17103403
2445:(1987).
2334:33730581
1823:f g
473:Examples
460:SINGULAR
449:SINGULAR
139:used in
65:May 2024
2814:Bibcode
2782:1086566
2744:0344230
2736:1970844
2684:0296079
2654:Bibcode
2628:0901394
2601:1943036
2559:0430304
2539:Bibcode
2516:1701608
2435:6955564
2427:2231202
2407:Bibcode
2366:0290097
2289:2747775
2098:, with
1976: (
435: (
174: (
166: (
127:, the
103: (
47:improve
45:Please
2832:
2780:
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2425:
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2322:
2287:
2277:
1962:has a
447:, and
170:) and
151:, and
107:) and
91:, the
36:, but
2830:S2CID
2804:arXiv
2732:JSTOR
2676:61638
2672:JSTOR
2563:S2CID
2431:S2CID
2397:arXiv
2370:S2CID
2330:S2CID
2302:arXiv
2285:S2CID
2257:arXiv
2205:Notes
1958:with
1845:) at
1662:to a
2698:PMID
2587:ISBN
2502:ISBN
2478:ISBN
2455:ISBN
2320:ISBN
2275:ISBN
2151:and
1997:CERN
1993:1997
1978:1999
1950:The
1342:are
895:then
550:then
467:2010
456:2009
437:2006
176:2003
168:1987
160:1995
117:1974
113:1972
105:1971
2822:doi
2800:389
2768:doi
2764:120
2724:doi
2720:100
2688:PMC
2662:doi
2547:doi
2415:doi
2393:142
2354:doi
2312:doi
2267:doi
1837:of
1809:If
1589:If
1523:det
1446:det
1333:If
794:If
478:If
186:If
119:),
87:In
2847::
2828:.
2820:.
2812:.
2798:.
2784:.
2778:MR
2776:.
2762:.
2758:.
2740:MR
2738:.
2730:.
2696:.
2686:.
2680:MR
2678:.
2670:.
2660:.
2650:69
2648:.
2642:.
2624:MR
2620:62
2618:.
2612:.
2597:MR
2595:.
2585:.
2561:.
2555:MR
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2545:.
2535:38
2533:.
2512:MR
2510:.
2476:.
2453:.
2429:.
2423:MR
2421:.
2413:.
2405:.
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2348:.
2328:.
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2273:.
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1097:so
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451:.
417:.
407:.
155:.
147:,
115:,
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2806::
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2656::
2630:.
2603:.
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2549::
2541::
2518:.
2486:.
2463:.
2437:.
2417::
2409::
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2110:P
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2027:x
2024:(
2019:1
2015:f
2011:(
1936:.
1933:)
1930:x
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1918:x
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963:(
960:f
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875:n
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814:=
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72:(
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63:(
42:.
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.