2699:
The subtraction of only multiples of 2 from the maximal number of positive roots occurs because the polynomial may have nonreal roots, which always come in pairs since the rule applies to polynomials whose coefficients are real. Thus if the polynomial is known to have all real roots, this rule allows
45:
by examining sign changes in its coefficients. The number of positive real roots is at most the number of sign changes in the sequence of polynomial's coefficients (omitting zero coefficients), and the difference between the root count and the sign change count is always even. In particular, when the
132:
of the rule, the number of negative roots is the number of sign changes after multiplying the coefficients of odd-power terms by â1, or fewer than it by an even number. This procedure is equivalent to substituting the negation of the variable for the variable itself. For example, the negative roots
2760:
that generalises
Descartes' rule. The rule of signs can be thought of as stating that the number of real roots of a polynomial is dependent on the polynomial's complexity, and that this complexity is proportional to the number of monomials it has, not its degree. KhovanskiÇ showed that this holds
61:. Repeated division of an interval in two results in a set of disjoint intervals, each containing one root, and together listing all the roots. This approach is used in the fastest algorithms today for computer computation of real roots of polynomials (see
1484:, so it must cross the positive x-axis an even number of times (each of which contributes an odd number of roots), and glance (without crossing) the positive x-axis an arbitrary number of times (each of which contributes an even number of roots).
2310:
429:. Therefore, it has exactly one positive root. To find the number of negative roots, change the signs of the coefficients of the terms with odd exponents, i.e., apply Descartes' rule of signs to the polynomial
336:
849:
784:
1482:
1742:
500:
2700:
one to find the exact number of positive and negative roots. Since it is easy to determine the multiplicity of zero as a root, the sign of all roots can be determined in this case.
675:
1966:
420:
588:
2165:
192:
2116:
1005:
2629:
2061:
2012:
1435:
1322:
1250:
2566:
2412:
1595:
882:
2679:
1130:
1164:
2489:
1911:
1800:
1768:
1676:
1197:
1650:
2368:
2339:
1551:
1522:
1387:
1351:
1279:
1039:
916:
719:
1886:
1860:
1840:
1820:
1618:
1093:
1073:
956:
936:
106:
of the polynomial is either equal to the number of sign changes between consecutive (nonzero) coefficients, or is less than it by an even number. A root of
2174:
2790:
2443:) is a polynomial with real coefficients which does not have a root at 0 (that is a polynomial with a nonzero constant term) then the
200:
508:, meaning that this second polynomial has two or zero positive roots; thus the original polynomial has two or zero negative roots.
341:
Thus, applying
Descartes' rule of signs to this polynomial gives the maximum number of negative roots of the original polynomial.
789:
120:
In particular, if the number of sign changes is zero or one, the number of positive roots equals the number of sign changes.
2957:
2895:
53:
of the variable makes it possible to use the rule of signs to count roots in any interval. This is the basic idea of
724:
50:
2634:
has no sign change, the original polynomial has no negative real roots. So the minimum number of nonreal roots is
1440:
2886:. Translations of Mathematical Monographs. Translated from the Russian by Smilka Zdravkovska. Providence, RI:
2502:
denotes the maximum number of negative roots (both of which can be found using
Descartes' rule of signs), and
2887:
432:
602:
512:
355:
521:
2684:
Since nonreal roots of a polynomial with real coefficients must occur in conjugate pairs, it means that
136:
961:
680:
So here, the roots are +1 (twice) and â1 (once), the negation of the roots of the original polynomial.
1916:
2577:
1974:
1392:
1284:
1212:
58:
2121:
1684:
2520:
2373:
1556:
2066:
2796:
2762:
854:
2640:
2017:
1102:
1135:
2453:
2784:
1747:
1655:
1169:
2787: â Relationship between the rational roots of a polynomial and its extreme coefficients
1623:
2905:
2344:
2315:
1527:
1498:
1363:
1327:
1255:
1015:
892:
695:
46:
number of sign changes is zero or one, then there are exactly zero or one positive roots.
8:
1865:
62:
2799: â Geometric relation between the roots of a polynomial and those of its derivative
1891:
1780:
2927:
2778:
1845:
1825:
1805:
1774:
1603:
1078:
1058:
941:
921:
103:
54:
38:
2891:
2838:
2766:
2751:
33:
425:
has one sign change between the second and third terms, as the sequence of signs is
28:
2901:
2830:
2941:
688:
The following is a rough outline of a proof. First, some preliminary definitions:
1166:, which would not change its number of strictly positive roots. Thus WLOG, let
2951:
2935:
2842:
2725:
2571:
has one sign change; so the maximum number of positive real roots is one. As
2432:
107:
78:
for using his rule for getting information of the number of negative roots.
102:
are ordered by descending variable exponent, then the number of positive
99:
96:
20:
2923:
92:
42:
2818:
2761:
true not just for polynomials but for algebraic combinations of many
2756:
129:
2922:
This article incorporates material from
Descartes' rule of signs on
2872:, Comptes Rendus Acad. Bulg. Sci. tome 63, No. 7, 2010, pp. 943â952.
2834:
2691:
has exactly two nonreal roots and one real root, which is positive.
2509:
2305:{\displaystyle Z(f)\leq Z(f')+1=V(f')-2s+1\leq V(f)-2s+1\leq V(f)+1}
504:
This polynomial has two sign changes, as the sequence of signs is
1055:
The number of strictly positive roots (counting multiplicity) of
1041:
be the number of strictly positive roots (counting multiplicity).
331:{\displaystyle a(-x)^{3}+b(-x)^{2}+c(-x)+d=-ax^{3}+bx^{2}-cx+d.}
91:
The rule states that if the nonzero terms of a single-variable
2716:
real positive roots counted with multiplicity, then for every
1075:
is equal to the number of sign changes in the coefficients of
1045:
With these, we can formally state
Descartes' rule as follows:
2859:, Annals of Mathematics., Vol. 19, No. 4, 1918, pp. 251â278.
844:{\displaystyle 0\leq b_{0}<b_{1}<\cdots <b_{n}}
2724:
changes of sign in the sequence of coefficients of the
2643:
2580:
2523:
2456:
2376:
2347:
2318:
2177:
2124:
2069:
2020:
1977:
1919:
1894:
1868:
1848:
1828:
1808:
1783:
1750:
1687:
1658:
1626:
1606:
1559:
1530:
1501:
1443:
1395:
1366:
1330:
1287:
1258:
1215:
1172:
1138:
1105:
1081:
1061:
1018:
964:
944:
924:
918:
be the number of sign changes of the coefficients of
895:
857:
792:
727:
698:
605:
524:
435:
358:
203:
139:
2793: â Geometry of the location of polynomial roots
16:
Counting polynomial real roots based on coefficients
2673:
2623:
2560:
2483:
2406:
2362:
2333:
2304:
2159:
2110:
2055:
2006:
1960:
1905:
1880:
1854:
1834:
1814:
1794:
1762:
1736:
1670:
1644:
1612:
1589:
1545:
1516:
1476:
1429:
1381:
1345:
1316:
1273:
1244:
1191:
1158:
1124:
1087:
1067:
1033:
999:
950:
930:
910:
876:
843:
778:
713:
669:
582:
494:
414:
330:
186:
2510:Example: some zero coefficients and nonreal roots
2949:
2928:Creative Commons Attribution/Share-Alike License
2870:A mapping defined by the SchurâSzegĆ composition
2781: â Counting polynomial roots in an interval
1553:always have the same parity. It remains to show
2498:denotes the maximum number of positive roots,
2435:, if counted according to multiplicity. So if
596:The factorization of the second polynomial is
2857:Recent extensions of Descartes' rule of signs
2819:"A Simple Proof of Descartes's Rule of Signs"
1777:, there exists at least one positive root of
779:{\displaystyle \sum _{i=0}^{n}a_{i}x^{b_{i}}}
1802:between any two different positive roots of
344:
593:so the roots are â1 (twice) and +1 (once).
81:
2881:
2791:Geometrical properties of polynomial roots
68:Descartes himself used the transformation
1477:{\displaystyle f(+\infty )=+\infty >0}
1132:, then we can divide the polynomial by
495:{\displaystyle f(-x)=-x^{3}+x^{2}+x-1.}
2950:
2506:denotes the degree of the polynomial.
670:{\displaystyle f(-x)=-(x-1)^{2}(x+1).}
415:{\displaystyle f(x)=+x^{3}+x^{2}-x-1}
2816:
2447:number of nonreal roots is equal to
583:{\displaystyle f(x)=(x+1)^{2}(x-1),}
1095:, minus a nonnegative even number.
13:
2703:
1465:
1453:
187:{\displaystyle ax^{3}+bx^{2}+cx+d}
14:
2969:
2916:
2823:The American Mathematical Monthly
2817:Wang, Xiaoshen (JuneâJuly 2004).
2427:th degree polynomial has exactly
1652:, then it is obvious. Now assume
1000:{\displaystyle a_{k}a_{k+1}<0}
123:
86:
2418:
1961:{\displaystyle Z(f')\geq Z(f)-1}
1495:From the lemma, it follows that
51:linear fractional transformation
2694:
2624:{\displaystyle f(-x)=-x^{3}-1,}
2007:{\displaystyle a_{0}a_{1}>0}
1430:{\displaystyle f(0)=a_{0}>0}
1317:{\displaystyle a_{0}a_{n}<0}
1245:{\displaystyle a_{n}a_{0}>0}
2926:, which is licensed under the
2875:
2862:
2849:
2810:
2662:
2650:
2593:
2584:
2533:
2527:
2475:
2463:
2401:
2395:
2386:
2380:
2370:have the same parity, we have
2357:
2351:
2328:
2322:
2293:
2287:
2263:
2257:
2233:
2222:
2207:
2196:
2187:
2181:
2160:{\displaystyle V(f')\leq V(f)}
2154:
2148:
2139:
2128:
2099:
2093:
2084:
2073:
2050:
2044:
2035:
2024:
1949:
1943:
1934:
1923:
1737:{\displaystyle Z(f')=V(f')-2s}
1722:
1711:
1702:
1691:
1584:
1578:
1569:
1563:
1540:
1534:
1511:
1505:
1456:
1447:
1405:
1399:
1376:
1370:
1340:
1334:
1268:
1262:
1028:
1022:
905:
899:
708:
702:
661:
649:
640:
627:
618:
609:
574:
562:
553:
540:
534:
528:
448:
439:
368:
362:
266:
257:
242:
232:
217:
207:
1:
2888:American Mathematical Society
2561:{\displaystyle f(x)=x^{3}-1,}
2407:{\displaystyle Z(f)\leq V(f)}
1590:{\displaystyle Z(f)\leq V(f)}
786:where we have integer powers
2111:{\displaystyle V(f')=V(f)-1}
7:
2772:
1842:-multiple positive root of
1487:The other case is similar.
877:{\displaystyle a_{i}\neq 0}
851:, and nonzero coefficients
515:of the first polynomial is
194:are the positive roots of
10:
2974:
2958:Theorems about polynomials
2739:). For sufficiently large
2720:> 0 there are at least
2674:{\displaystyle 3-(1+0)=2.}
2056:{\displaystyle V(f')=V(f)}
1125:{\displaystyle b_{0}>0}
2882:KhovanskiÇ, A.G. (1991).
1681:By induction hypothesis,
1159:{\displaystyle x^{b_{0}}}
345:Example: cubic polynomial
2942:Descartes' Rule of Signs
2936:Descartes' Rule of Signs
2803:
2763:transcendental functions
2754:developed the theory of
2484:{\displaystyle n-(p+q),}
938:, meaning the number of
683:
82:Descartes' rule of signs
25:Descartes' rule of signs
2708:If the real polynomial
1763:{\displaystyle s\geq 0}
1671:{\displaystyle n\geq 2}
1192:{\displaystyle b_{0}=0}
2747:such changes of sign.
2675:
2625:
2562:
2485:
2408:
2364:
2335:
2306:
2161:
2112:
2057:
2008:
1962:
1907:
1882:
1856:
1836:
1816:
1796:
1764:
1738:
1672:
1646:
1614:
1591:
1547:
1518:
1478:
1431:
1383:
1347:
1318:
1275:
1246:
1193:
1160:
1126:
1089:
1069:
1035:
1001:
952:
932:
912:
878:
845:
780:
748:
715:
671:
584:
496:
416:
332:
188:
2785:Rational root theorem
2676:
2626:
2563:
2486:
2409:
2365:
2336:
2307:
2162:
2113:
2058:
2009:
1963:
1908:
1883:
1857:
1837:
1817:
1797:
1765:
1739:
1673:
1647:
1645:{\displaystyle n=0,1}
1615:
1592:
1548:
1519:
1492:Proof of main theorem
1479:
1432:
1384:
1348:
1319:
1276:
1247:
1194:
1161:
1127:
1090:
1070:
1036:
1002:
953:
933:
913:
879:
846:
781:
728:
716:
692:Write the polynomial
672:
585:
497:
417:
333:
189:
59:BudanâFourier theorem
2868:Vladimir P. Kostov,
2743:, there are exactly
2641:
2578:
2521:
2454:
2374:
2363:{\displaystyle V(f)}
2345:
2334:{\displaystyle Z(f)}
2316:
2175:
2122:
2067:
2018:
1975:
1917:
1892:
1866:
1846:
1826:
1806:
1781:
1748:
1685:
1656:
1624:
1604:
1557:
1546:{\displaystyle V(f)}
1528:
1517:{\displaystyle Z(f)}
1499:
1441:
1393:
1382:{\displaystyle f(x)}
1364:
1346:{\displaystyle Z(f)}
1328:
1285:
1274:{\displaystyle Z(f)}
1256:
1213:
1170:
1136:
1103:
1079:
1059:
1034:{\displaystyle Z(f)}
1016:
962:
942:
922:
911:{\displaystyle V(f)}
893:
855:
790:
725:
714:{\displaystyle f(x)}
696:
603:
522:
433:
356:
201:
137:
2944:â Basic explanation
2938:â Proof of the rule
2797:GaussâLucas theorem
2169:Together, we have
1881:{\displaystyle k-1}
1207: —
1053: —
63:real-root isolation
2767:Pfaffian functions
2671:
2621:
2558:
2481:
2404:
2360:
2331:
2302:
2157:
2108:
2053:
2004:
1958:
1906:{\displaystyle f'}
1903:
1888:-multiple root of
1878:
1852:
1832:
1812:
1795:{\displaystyle f'}
1792:
1760:
1734:
1668:
1642:
1610:
1587:
1543:
1514:
1493:
1474:
1427:
1379:
1359:
1343:
1314:
1271:
1242:
1205:
1189:
1156:
1122:
1085:
1065:
1051:
1031:
997:
948:
928:
908:
874:
841:
776:
711:
667:
580:
492:
412:
328:
184:
2752:Askold Khovanskii
2118:. In both cases,
1855:{\displaystyle f}
1835:{\displaystyle k}
1815:{\displaystyle f}
1744:for some integer
1613:{\displaystyle n}
1491:
1357:
1203:
1088:{\displaystyle f}
1068:{\displaystyle f}
1049:
951:{\displaystyle k}
931:{\displaystyle f}
2965:
2910:
2909:
2879:
2873:
2866:
2860:
2853:
2847:
2846:
2814:
2765:, the so-called
2728:of the function
2690:
2680:
2678:
2677:
2672:
2630:
2628:
2627:
2622:
2611:
2610:
2567:
2565:
2564:
2559:
2548:
2547:
2490:
2488:
2487:
2482:
2413:
2411:
2410:
2405:
2369:
2367:
2366:
2361:
2340:
2338:
2337:
2332:
2311:
2309:
2308:
2303:
2232:
2206:
2166:
2164:
2163:
2158:
2138:
2117:
2115:
2114:
2109:
2083:
2062:
2060:
2059:
2054:
2034:
2013:
2011:
2010:
2005:
1997:
1996:
1987:
1986:
1967:
1965:
1964:
1959:
1933:
1912:
1910:
1909:
1904:
1902:
1887:
1885:
1884:
1879:
1861:
1859:
1858:
1853:
1841:
1839:
1838:
1833:
1821:
1819:
1818:
1813:
1801:
1799:
1798:
1793:
1791:
1769:
1767:
1766:
1761:
1743:
1741:
1740:
1735:
1721:
1701:
1677:
1675:
1674:
1669:
1651:
1649:
1648:
1643:
1619:
1617:
1616:
1611:
1596:
1594:
1593:
1588:
1552:
1550:
1549:
1544:
1523:
1521:
1520:
1515:
1483:
1481:
1480:
1475:
1436:
1434:
1433:
1428:
1420:
1419:
1388:
1386:
1385:
1380:
1352:
1350:
1349:
1344:
1323:
1321:
1320:
1315:
1307:
1306:
1297:
1296:
1280:
1278:
1277:
1272:
1251:
1249:
1248:
1243:
1235:
1234:
1225:
1224:
1208:
1198:
1196:
1195:
1190:
1182:
1181:
1165:
1163:
1162:
1157:
1155:
1154:
1153:
1152:
1131:
1129:
1128:
1123:
1115:
1114:
1094:
1092:
1091:
1086:
1074:
1072:
1071:
1066:
1054:
1040:
1038:
1037:
1032:
1006:
1004:
1003:
998:
990:
989:
974:
973:
957:
955:
954:
949:
937:
935:
934:
929:
917:
915:
914:
909:
883:
881:
880:
875:
867:
866:
850:
848:
847:
842:
840:
839:
821:
820:
808:
807:
785:
783:
782:
777:
775:
774:
773:
772:
758:
757:
747:
742:
720:
718:
717:
712:
676:
674:
673:
668:
648:
647:
589:
587:
586:
581:
561:
560:
507:
501:
499:
498:
493:
479:
478:
466:
465:
428:
421:
419:
418:
413:
399:
398:
386:
385:
337:
335:
334:
329:
309:
308:
293:
292:
250:
249:
225:
224:
193:
191:
190:
185:
168:
167:
152:
151:
116:
112:
77:
2973:
2972:
2968:
2967:
2966:
2964:
2963:
2962:
2948:
2947:
2919:
2914:
2913:
2898:
2880:
2876:
2867:
2863:
2855:D. R. Curtiss,
2854:
2850:
2835:10.2307/4145072
2815:
2811:
2806:
2779:Sturm's theorem
2775:
2706:
2704:Generalizations
2697:
2685:
2642:
2639:
2638:
2606:
2602:
2579:
2576:
2575:
2543:
2539:
2522:
2519:
2518:
2514:The polynomial
2512:
2455:
2452:
2451:
2421:
2416:
2375:
2372:
2371:
2346:
2343:
2342:
2317:
2314:
2313:
2312:Further, since
2225:
2199:
2176:
2173:
2172:
2131:
2123:
2120:
2119:
2076:
2068:
2065:
2064:
2027:
2019:
2016:
2015:
1992:
1988:
1982:
1978:
1976:
1973:
1972:
1926:
1918:
1915:
1914:
1895:
1893:
1890:
1889:
1867:
1864:
1863:
1847:
1844:
1843:
1827:
1824:
1823:
1807:
1804:
1803:
1784:
1782:
1779:
1778:
1775:Rolle's theorem
1749:
1746:
1745:
1714:
1694:
1686:
1683:
1682:
1657:
1654:
1653:
1625:
1622:
1621:
1605:
1602:
1601:
1558:
1555:
1554:
1529:
1526:
1525:
1500:
1497:
1496:
1489:
1442:
1439:
1438:
1415:
1411:
1394:
1391:
1390:
1365:
1362:
1361:
1355:
1329:
1326:
1325:
1302:
1298:
1292:
1288:
1286:
1283:
1282:
1257:
1254:
1253:
1230:
1226:
1220:
1216:
1214:
1211:
1210:
1206:
1177:
1173:
1171:
1168:
1167:
1148:
1144:
1143:
1139:
1137:
1134:
1133:
1110:
1106:
1104:
1101:
1100:
1097:
1080:
1077:
1076:
1060:
1057:
1056:
1052:
1017:
1014:
1013:
979:
975:
969:
965:
963:
960:
959:
943:
940:
939:
923:
920:
919:
894:
891:
890:
862:
858:
856:
853:
852:
835:
831:
816:
812:
803:
799:
791:
788:
787:
768:
764:
763:
759:
753:
749:
743:
732:
726:
723:
722:
697:
694:
693:
686:
643:
639:
604:
601:
600:
556:
552:
523:
520:
519:
505:
474:
470:
461:
457:
434:
431:
430:
426:
394:
390:
381:
377:
357:
354:
353:
349:The polynomial
347:
304:
300:
288:
284:
245:
241:
220:
216:
202:
199:
198:
163:
159:
147:
143:
138:
135:
134:
126:
114:
110:
89:
84:
69:
55:Budan's theorem
27:, described by
17:
12:
11:
5:
2971:
2961:
2960:
2946:
2945:
2939:
2918:
2917:External links
2915:
2912:
2911:
2896:
2890:. p. 88.
2874:
2861:
2848:
2808:
2807:
2805:
2802:
2801:
2800:
2794:
2788:
2782:
2774:
2771:
2705:
2702:
2696:
2693:
2682:
2681:
2670:
2667:
2664:
2661:
2658:
2655:
2652:
2649:
2646:
2632:
2631:
2620:
2617:
2614:
2609:
2605:
2601:
2598:
2595:
2592:
2589:
2586:
2583:
2569:
2568:
2557:
2554:
2551:
2546:
2542:
2538:
2535:
2532:
2529:
2526:
2511:
2508:
2492:
2491:
2480:
2477:
2474:
2471:
2468:
2465:
2462:
2459:
2446:
2420:
2417:
2403:
2400:
2397:
2394:
2391:
2388:
2385:
2382:
2379:
2359:
2356:
2353:
2350:
2330:
2327:
2324:
2321:
2301:
2298:
2295:
2292:
2289:
2286:
2283:
2280:
2277:
2274:
2271:
2268:
2265:
2262:
2259:
2256:
2253:
2250:
2247:
2244:
2241:
2238:
2235:
2231:
2228:
2224:
2221:
2218:
2215:
2212:
2209:
2205:
2202:
2198:
2195:
2192:
2189:
2186:
2183:
2180:
2156:
2153:
2150:
2147:
2144:
2141:
2137:
2134:
2130:
2127:
2107:
2104:
2101:
2098:
2095:
2092:
2089:
2086:
2082:
2079:
2075:
2072:
2052:
2049:
2046:
2043:
2040:
2037:
2033:
2030:
2026:
2023:
2003:
2000:
1995:
1991:
1985:
1981:
1957:
1954:
1951:
1948:
1945:
1942:
1939:
1936:
1932:
1929:
1925:
1922:
1901:
1898:
1877:
1874:
1871:
1851:
1831:
1811:
1790:
1787:
1759:
1756:
1753:
1733:
1730:
1727:
1724:
1720:
1717:
1713:
1710:
1707:
1704:
1700:
1697:
1693:
1690:
1667:
1664:
1661:
1641:
1638:
1635:
1632:
1629:
1609:
1586:
1583:
1580:
1577:
1574:
1571:
1568:
1565:
1562:
1542:
1539:
1536:
1533:
1513:
1510:
1507:
1504:
1490:
1473:
1470:
1467:
1464:
1461:
1458:
1455:
1452:
1449:
1446:
1426:
1423:
1418:
1414:
1410:
1407:
1404:
1401:
1398:
1378:
1375:
1372:
1369:
1358:Proof of Lemma
1356:
1342:
1339:
1336:
1333:
1313:
1310:
1305:
1301:
1295:
1291:
1270:
1267:
1264:
1261:
1241:
1238:
1233:
1229:
1223:
1219:
1201:
1188:
1185:
1180:
1176:
1151:
1147:
1142:
1121:
1118:
1113:
1109:
1084:
1064:
1047:
1043:
1042:
1030:
1027:
1024:
1021:
1009:
1008:
996:
993:
988:
985:
982:
978:
972:
968:
947:
927:
907:
904:
901:
898:
886:
885:
873:
870:
865:
861:
838:
834:
830:
827:
824:
819:
815:
811:
806:
802:
798:
795:
771:
767:
762:
756:
752:
746:
741:
738:
735:
731:
710:
707:
704:
701:
685:
682:
678:
677:
666:
663:
660:
657:
654:
651:
646:
642:
638:
635:
632:
629:
626:
623:
620:
617:
614:
611:
608:
591:
590:
579:
576:
573:
570:
567:
564:
559:
555:
551:
548:
545:
542:
539:
536:
533:
530:
527:
491:
488:
485:
482:
477:
473:
469:
464:
460:
456:
453:
450:
447:
444:
441:
438:
423:
422:
411:
408:
405:
402:
397:
393:
389:
384:
380:
376:
373:
370:
367:
364:
361:
346:
343:
339:
338:
327:
324:
321:
318:
315:
312:
307:
303:
299:
296:
291:
287:
283:
280:
277:
274:
271:
268:
265:
262:
259:
256:
253:
248:
244:
240:
237:
234:
231:
228:
223:
219:
215:
212:
209:
206:
183:
180:
177:
174:
171:
166:
162:
158:
155:
150:
146:
142:
125:
124:Negative roots
122:
113:is counted as
88:
87:Positive roots
85:
83:
80:
29:René Descartes
15:
9:
6:
4:
3:
2:
2970:
2959:
2956:
2955:
2953:
2943:
2940:
2937:
2934:
2933:
2932:
2931:
2929:
2925:
2907:
2903:
2899:
2897:0-8218-4547-0
2893:
2889:
2885:
2878:
2871:
2865:
2858:
2852:
2844:
2840:
2836:
2832:
2828:
2824:
2820:
2813:
2809:
2798:
2795:
2792:
2789:
2786:
2783:
2780:
2777:
2776:
2770:
2768:
2764:
2759:
2758:
2753:
2750:In the 1970s
2748:
2746:
2742:
2738:
2734:
2731:
2727:
2726:Taylor series
2723:
2719:
2715:
2711:
2701:
2692:
2688:
2668:
2665:
2659:
2656:
2653:
2647:
2644:
2637:
2636:
2635:
2618:
2615:
2612:
2607:
2603:
2599:
2596:
2590:
2587:
2581:
2574:
2573:
2572:
2555:
2552:
2549:
2544:
2540:
2536:
2530:
2524:
2517:
2516:
2515:
2507:
2505:
2501:
2497:
2478:
2472:
2469:
2466:
2460:
2457:
2450:
2449:
2448:
2444:
2442:
2438:
2434:
2433:complex plane
2431:roots in the
2430:
2426:
2419:Nonreal roots
2415:
2398:
2392:
2389:
2383:
2377:
2354:
2348:
2325:
2319:
2299:
2296:
2290:
2284:
2281:
2278:
2275:
2272:
2269:
2266:
2260:
2254:
2251:
2248:
2245:
2242:
2239:
2236:
2229:
2226:
2219:
2216:
2213:
2210:
2203:
2200:
2193:
2190:
2184:
2178:
2170:
2167:
2151:
2145:
2142:
2135:
2132:
2125:
2105:
2102:
2096:
2090:
2087:
2080:
2077:
2070:
2047:
2041:
2038:
2031:
2028:
2021:
2001:
1998:
1993:
1989:
1983:
1979:
1969:
1955:
1952:
1946:
1940:
1937:
1930:
1927:
1920:
1899:
1896:
1875:
1872:
1869:
1849:
1829:
1809:
1788:
1785:
1776:
1771:
1757:
1754:
1751:
1731:
1728:
1725:
1718:
1715:
1708:
1705:
1698:
1695:
1688:
1679:
1665:
1662:
1659:
1639:
1636:
1633:
1630:
1627:
1607:
1600:We induct on
1598:
1581:
1575:
1572:
1566:
1560:
1537:
1531:
1508:
1502:
1488:
1485:
1471:
1468:
1462:
1459:
1450:
1444:
1424:
1421:
1416:
1412:
1408:
1402:
1396:
1373:
1367:
1354:
1337:
1331:
1311:
1308:
1303:
1299:
1293:
1289:
1265:
1259:
1239:
1236:
1231:
1227:
1221:
1217:
1200:
1186:
1183:
1178:
1174:
1149:
1145:
1140:
1119:
1116:
1111:
1107:
1096:
1082:
1062:
1046:
1025:
1019:
1011:
1010:
994:
991:
986:
983:
980:
976:
970:
966:
945:
925:
902:
896:
888:
887:
871:
868:
863:
859:
836:
832:
828:
825:
822:
817:
813:
809:
804:
800:
796:
793:
769:
765:
760:
754:
750:
744:
739:
736:
733:
729:
705:
699:
691:
690:
689:
681:
664:
658:
655:
652:
644:
636:
633:
630:
624:
621:
615:
612:
606:
599:
598:
597:
594:
577:
571:
568:
565:
557:
549:
546:
543:
537:
531:
525:
518:
517:
516:
514:
513:factorization
511:In fact, the
509:
502:
489:
486:
483:
480:
475:
471:
467:
462:
458:
454:
451:
445:
442:
436:
409:
406:
403:
400:
395:
391:
387:
382:
378:
374:
371:
365:
359:
352:
351:
350:
342:
325:
322:
319:
316:
313:
310:
305:
301:
297:
294:
289:
285:
281:
278:
275:
272:
269:
263:
260:
254:
251:
246:
238:
235:
229:
226:
221:
213:
210:
204:
197:
196:
195:
181:
178:
175:
172:
169:
164:
160:
156:
153:
148:
144:
140:
131:
121:
118:
109:
105:
101:
98:
94:
79:
76:
72:
66:
64:
60:
56:
52:
47:
44:
40:
37:, counts the
36:
35:
30:
26:
22:
2921:
2920:
2883:
2877:
2869:
2864:
2856:
2851:
2826:
2822:
2812:
2755:
2749:
2744:
2740:
2736:
2732:
2729:
2721:
2717:
2713:
2709:
2707:
2698:
2695:Special case
2686:
2683:
2633:
2570:
2513:
2503:
2499:
2495:
2493:
2440:
2436:
2428:
2424:
2422:
2171:
2168:
1970:
1822:. Also, any
1772:
1680:
1599:
1494:
1486:
1437:and ends at
1360:
1281:is even. If
1202:
1098:
1048:
1044:
687:
679:
595:
592:
510:
506:(â, +, +, â)
503:
427:(+, +, â, â)
424:
348:
340:
127:
119:
108:multiplicity
100:coefficients
90:
74:
70:
67:
48:
34:La Géométrie
32:
24:
18:
21:mathematics
2924:PlanetMath
2906:0728.12002
2884:Fewnomials
2829:(6): 525.
2757:fewnomials
1389:starts at
958:such that
93:polynomial
43:polynomial
2843:0002-9890
2648:−
2613:−
2600:−
2588:−
2550:−
2461:−
2390:≤
2282:≤
2267:−
2252:≤
2237:−
2191:≤
2143:≤
2103:−
1953:−
1938:≥
1873:−
1755:≥
1726:−
1663:≥
1573:≤
1466:∞
1454:∞
869:≠
826:⋯
797:≤
730:∑
634:−
625:−
613:−
569:−
487:−
455:−
443:−
407:−
401:−
311:−
279:−
261:−
236:−
211:−
130:corollary
2952:Category
2773:See also
2230:′
2204:′
2136:′
2081:′
2032:′
1931:′
1900:′
1789:′
1719:′
1699:′
1353:is odd.
57:and the
2445:minimum
2063:, else
2014:, then
1913:. Thus
1324:, then
1252:, then
1050:Theorem
117:roots.
31:in his
2904:
2894:
2841:
2494:where
2804:Notes
1862:is a
1620:. If
1204:Lemma
684:Proof
128:As a
104:roots
95:with
41:of a
39:roots
2892:ISBN
2839:ISSN
2712:has
2423:Any
2341:and
1999:>
1524:and
1469:>
1422:>
1309:<
1237:>
1117:>
1012:Let
992:<
889:Let
829:<
823:<
810:<
97:real
2902:Zbl
2831:doi
2827:111
2689:â 1
1971:If
1773:By
1209:If
1099:If
721:as
133:of
73:â â
65:).
19:In
2954::
2900:.
2837:.
2825:.
2821:.
2769:.
2669:2.
2414:.
1968:.
1770:.
1678:.
1597:.
1199:.
490:1.
49:A
23:,
2930:.
2908:.
2845:.
2833::
2745:k
2741:a
2737:x
2735:(
2733:P
2730:e
2722:k
2718:a
2714:k
2710:P
2687:x
2666:=
2663:)
2660:0
2657:+
2654:1
2651:(
2645:3
2619:,
2616:1
2608:3
2604:x
2597:=
2594:)
2591:x
2585:(
2582:f
2556:,
2553:1
2545:3
2541:x
2537:=
2534:)
2531:x
2528:(
2525:f
2504:n
2500:q
2496:p
2479:,
2476:)
2473:q
2470:+
2467:p
2464:(
2458:n
2441:x
2439:(
2437:f
2429:n
2425:n
2402:)
2399:f
2396:(
2393:V
2387:)
2384:f
2381:(
2378:Z
2358:)
2355:f
2352:(
2349:V
2329:)
2326:f
2323:(
2320:Z
2300:1
2297:+
2294:)
2291:f
2288:(
2285:V
2279:1
2276:+
2273:s
2270:2
2264:)
2261:f
2258:(
2255:V
2249:1
2246:+
2243:s
2240:2
2234:)
2227:f
2223:(
2220:V
2217:=
2214:1
2211:+
2208:)
2201:f
2197:(
2194:Z
2188:)
2185:f
2182:(
2179:Z
2155:)
2152:f
2149:(
2146:V
2140:)
2133:f
2129:(
2126:V
2106:1
2100:)
2097:f
2094:(
2091:V
2088:=
2085:)
2078:f
2074:(
2071:V
2051:)
2048:f
2045:(
2042:V
2039:=
2036:)
2029:f
2025:(
2022:V
2002:0
1994:1
1990:a
1984:0
1980:a
1956:1
1950:)
1947:f
1944:(
1941:Z
1935:)
1928:f
1924:(
1921:Z
1897:f
1876:1
1870:k
1850:f
1830:k
1810:f
1786:f
1758:0
1752:s
1732:s
1729:2
1723:)
1716:f
1712:(
1709:V
1706:=
1703:)
1696:f
1692:(
1689:Z
1666:2
1660:n
1640:1
1637:,
1634:0
1631:=
1628:n
1608:n
1585:)
1582:f
1579:(
1576:V
1570:)
1567:f
1564:(
1561:Z
1541:)
1538:f
1535:(
1532:V
1512:)
1509:f
1506:(
1503:Z
1472:0
1463:+
1460:=
1457:)
1451:+
1448:(
1445:f
1425:0
1417:0
1413:a
1409:=
1406:)
1403:0
1400:(
1397:f
1377:)
1374:x
1371:(
1368:f
1341:)
1338:f
1335:(
1332:Z
1312:0
1304:n
1300:a
1294:0
1290:a
1269:)
1266:f
1263:(
1260:Z
1240:0
1232:0
1228:a
1222:n
1218:a
1187:0
1184:=
1179:0
1175:b
1150:0
1146:b
1141:x
1120:0
1112:0
1108:b
1083:f
1063:f
1029:)
1026:f
1023:(
1020:Z
1007:.
995:0
987:1
984:+
981:k
977:a
971:k
967:a
946:k
926:f
906:)
903:f
900:(
897:V
884:.
872:0
864:i
860:a
837:n
833:b
818:1
814:b
805:0
801:b
794:0
770:i
766:b
761:x
755:i
751:a
745:n
740:0
737:=
734:i
709:)
706:x
703:(
700:f
665:.
662:)
659:1
656:+
653:x
650:(
645:2
641:)
637:1
631:x
628:(
622:=
619:)
616:x
610:(
607:f
578:,
575:)
572:1
566:x
563:(
558:2
554:)
550:1
547:+
544:x
541:(
538:=
535:)
532:x
529:(
526:f
484:x
481:+
476:2
472:x
468:+
463:3
459:x
452:=
449:)
446:x
440:(
437:f
410:1
404:x
396:2
392:x
388:+
383:3
379:x
375:+
372:=
369:)
366:x
363:(
360:f
326:.
323:d
320:+
317:x
314:c
306:2
302:x
298:b
295:+
290:3
286:x
282:a
276:=
273:d
270:+
267:)
264:x
258:(
255:c
252:+
247:2
243:)
239:x
233:(
230:b
227:+
222:3
218:)
214:x
208:(
205:a
182:d
179:+
176:x
173:c
170:+
165:2
161:x
157:b
154:+
149:3
145:x
141:a
115:k
111:k
75:x
71:x
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