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1561:). The joys of vacuously true statements on the empty set! I think the only way to resolve this conflict is to set the degree of the polynomial zero to be either "undefined" or explicitly defined as some negative number (since the empty set is contained in every one of these degree "sets", it must be strictly smaller than all of them). -
484:. Our polynomials should not "shrink" when we multiply them. If f(x)=0, then f(x)*g(x)=0, no matter what degree g(x) has. This will break many of our theorems. Therefore, we do not define the degree at 0. (This is a special case of the "norm" function for a euclidean domain which requires that deg(f*g)≥deg(f) and deg(f*g)≥deg(g).)
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norm that satisfies the appropriate axioms. Finding such a norm is often very difficult, and there are often (perhaps always??) numerous norms whenever there is one (for example, if deg(f) is defined as we would expect and then we create a new function D, defined as D(f) = 42*deg(f), this is a valid
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The progress of the theory of stability of various types of functional equations such as quadratic, cubic, quartic, quintic, sextic, septic, octic, nonic, decic, undecic, duodecic, tredecic, quattordecic have been dealt by many mathematicians and there are lot of interesting and significant results
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I think 'hectic' is supposed to be math humor. We'll use 'bajillionic' if we really want, until the 100th degree polynomial actually appears in regular contexts, and then we'll name it appropriately. Sure, not everything gets a name (see 'once', 'twice', 'thrice', 4-ce? quince? ...), but those that
204:
Please, when you post a comment which does not belongs to an existing thread, put it in a new section at the end of the page. The "New section" button at the top of the talk page allow to do this easily. I so not understand you post: The first paragraph says correctly "The degree of a polynomial is
1910:
The example given in parentheses contradicts the statement made. If you're in an integral domain in the example, you can take the highest order term in each factor, and add them up. In terms of total number of operations, this is much "easier" than multiplying it all out first. For more general
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Generally a polynomial is defined as a sequence of elements (called coefficients) from a ring indexed by natural numbers with the following characteristic: there is a natural number n (called the degree) so that the n-th coefficient is non-zero and all coefficients with index higher than are zero.
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The wording to which you refer (which I inserted) is not clear and needs to be changed. What is being referred to is the behaviour of the degree of polynomials from the previous section: that the sum or difference of two polynomials has a degree that is less than or equal to the largest of the
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rings, you have to look for zero divisors and so on, but it's not obvious whether or not there is a more efficient method for doing this than doing the entire expansion. In any case, unless there is a definite computational idea in mind, "it is easier" sounds more like an opinion than a fact.
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has removed two times from the article that -1 is a common convention for the degree of 0. His argument is that he finds this convention not convenient. I agree with him that this convention breaks the formula for the degree of a product. However, the problem is not there. The article asserts
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For that matter, why there is a name for them at all? Even in the rare circumstance of one actually appearing, it would probably just get named "100th degree polynomial". They don't deserve a name any more than (say) 73rd degree ones do (and much less than 11th)...
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Another thing that may confuse you is that I define the polynomial as a function, instead of as a sequence, but that is equivalent. I find this way of doing things more elegant because you don't have bounds on the size of the sequence appearing anywhere.
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The thing is that the definition of the degree is embedded in the definition of the polynomial. What would be the best way to include here the formal/mathematically correct definition of the degree - should we repeat the polynomial definition here? -
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Aha.. I thought the function P was evaluating the polynomial in the ring, and I didn't understand what N was (now I see it). OK, now that I understand the setup, notice that even in your definition, for the zero polynomial, "the highest n such that
711:: I have noted your new section on abstract algebra. This is a good idea but as said above it should be based on formal definitions and united with the other articles. In particular, the last sentence is misleading. deg(0) is not undefined
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The degree of f(x)=c is zero, but only if c is not 0. The degree of 0 is undefined (but many say that the degree is minus infinity so that it "sorts" below the constants). Here are a couple justifications for this statement:
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1907:"To determine the degree of a polynomial that is not in standard form (for example (y − 3)(2y + 6)( − 4y − 21)) it is easier to expand or express the polynomial into a sum or difference of terms;"
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norm). So, I say that degree is not defined at 0 because degree is a norm and norms are not defined at 0. But, it is something of a circular argument, so there probably is a better way to state it.
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Two conflicting definitions seem to be given. The first paragraph says the degree is the sum of the powers of all terms, the second says it is the sum of powers in one term. That is confusing.
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We should have these formal definitions somewhere, and then everything follows, that R is a ring, that deg is a valuation, etc... Right now there are bits and pieces spread among the
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As far as your way of defining a polynomial, I'm not following. As I read your definition, if we let R be the real numbers, then how is exp(x) not a polynomial by your definition? -
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More formally: a polynomial is a function P from N (natural numbers) to R (where R is a ring) where there is n in N (the degree) so that P(n) is not zero and for all m: -->
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in a general context. What is important is whether it is used by convention; here mention of both conventions appears to be appropriate, but the explanation shows why
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would be 3?! I'm not a mathematician so I'm a bit reluctant to change the article, but would it be reasonable to clarify this in the article? Or maybe I was wrong? --
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Why is x²+xy+y² called a binomial based on having two variables and not a trinomial based on have three terms (in two variables). If you look at the article
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represents the zero polynomial, we can conclude that the degree of the zero polynomial is any arbitrary natural number (with the proper choice of P
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Here is a mathematics book that defines the degree of the zero polynomial to be -1: "By convention, the zero polynomial has degree -1." (p. 233)
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degree of each of the polynomials, and the product of two polynomials has a degree that is the sum of the degrees of each of the polynomials. —
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Ease of writing programs is no reason for a mathematical convention: it is only a reason for a choice of representation by a programmer, where
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That is, by implication, what is already being stated. But you put it more clearly; cleaning up the presentation in the article would help. —
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is a function from non-zero polynomials to non-negative integers, so the section on "Behaviour" might be better called "Properties of the
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degree of a function being "D", can |D|<1, while squreroots are 1/2 i'm not very sure, on this. but it isn't adressed in the artical
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I'm new to this
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x²+xy+y² is nowhere called a binomial. It is a trinomial. "Binary" and "binomial" are two different words that must not be confused.
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The point is that there is no formal (mathematically correct) definition given. What you have is just an explanation/illustration.
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the norm is undefined at zero, it is undefined due to the way deg was defined. Deriving stuff in the proper order is important.
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function". A problem with the "Behaviour" section is that it doesn't say that the zero polynomial is not in the domain of the
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Yes. The question is now how to best integrate these formal definitions (or equivalent) in the various articles I mentioned.
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Perhaps you should read more carefully; I wrote above: "Obviously this would then not be defined for the zero polynomial".
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is an empty set. What would the maximum (or even supremum) of an empty set look like? You could make a theorem that if
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I've had a bit of a debate with someone regarding degrees 0 and 1 of a polynomial. My argument is this: the degree of
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I does make sense, I have now figured it out. I guess the proper way to define this would go along those lines:
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i.e. it is something like P={a0,a1,...,an,0,0,...} where a0,...an are ellements of R and 0 is the zero in R.
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function. With that in mind, the section on the degree of the zero polynomial can say that the domain of
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that mathematicians regard -1 as a poor choice, I have cited both editions in the article. Do you have a
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do are named humorously at first (see 'velocity', 'acceleration', 'jerk', 'snap', 'crackle', and 'pop'.)
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Your are confused by the distinction between a polynomial (an abstract object) and a polynomial function.
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The degree = the highest power of X with a non-zero coeff. Indeed, the degree of the polynomial
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the third example given is 3ts+9t+5s -- which has two variables and is not called a binomial.
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i is the squart root of -1. So can someone explain what degree i and its factors are? --
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is injective, but it is important to understand that they are different objects.
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Childs uses −∞ and explains the advantages of using −∞ (p. 288). Since this is
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Hectic links to itself - the link should be removed or a stub should be made!
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So you did, my mistake. So, I guess we really are in agreement on all this? -
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is not a polynomial, and this article is about the "Degree of a polynomial".
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I'm not sure if this is enough reasoning for you... Does this make sense? -
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Excellent! Thank you. I see now I misread and didn't distinguish the two!
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above is correct in the sense that it doesn't exclude the zero polynomial.
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can be extended to include the zero polynomial by defining deg 0 = −∞. --
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1866:. That said, they seem to get quite rare after 10, certainly after 12.
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the highest degree of its terms". "Highest" is not the same as "sum".
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gives the coefficient of the nth power of the formal variable
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is not a polynomial function, because there is no polynomial
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The numbers beyond 10 can occasionally be encountered, e.g.
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I think the article covers this all perfectly adequately. —
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Note that this definition, contrary to the one proposed by
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I have corrected the statement and changed the example.
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This is completely bogus. The degree of f(x) = 0 is 0.
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that explains why -1 is preferable in some cases? --
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articles. Any idea on how to unify these articles ?
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The degree of the zero polynomial is minus infinity
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1338:
1315:
1307:
1289:
1287:
1286:
1281:
1276:
1238:
1236:
1235:
1230:
1216:
1215:
1199:
1197:
1196:
1191:
1189:
1177:
1175:
1174:
1169:
1157:
1155:
1154:
1149:
1147:
1142:
1137:
1113:
1111:
1110:
1105:
1103:
1102:
1093:
1078:
1076:
1075:
1070:
1068:
1056:
1054:
1053:
1048:
1046:
1034:
1032:
1031:
1026:
1024:
1023:
1007:
1005:
1004:
999:
987:
985:
984:
979:
977:
976:
975:
952:
948:
931:
926:
898:
897:
881:
879:
878:
873:
868:
857:
856:
840:
838:
837:
832:
820:
818:
817:
812:
801:To a polynomial
797:
795:
794:
789:
777:
775:
774:
769:
691:
689:
688:
683:
678:
654:
652:
651:
646:
621:
619:
618:
613:
590:
585:
564:
562:
561:
556:
551:
546:
528:
526:
525:
520:
459:Euclidean domain
392:
390:
389:
384:
382:
381:
340:
338:
337:
332:
330:
329:
310:
308:
307:
302:
300:
299:
277:
275:
274:
269:
267:
266:
247:
245:
244:
239:
123:
122:
119:
116:
113:
92:
87:
86:
76:
69:
68:
63:
55:
48:
31:
25:
24:
16:
2417:
2416:
2412:
2411:
2410:
2408:
2407:
2406:
2387:
2386:
2335:
2305:
2181:
2180:
2170:
2120:
2076:I agree, but 1/
2046:
2042:
2038:
2034:
2017:reliable source
2013:strong evidence
1949:
1912:
1905:
1886:
1802:preferred over
1796:
1778:
1729:
1728:
1695:
1679:
1663:
1649:
1648:
1629:
1628:
1616:
1614:
1611:
1560:
1556:
1515:
1480:
1463:
1462:
1423:
1373:
1349:
1348:
1292:
1291:
1255:
1254:
1207:
1202:
1201:
1180:
1179:
1160:
1159:
1116:
1115:
1094:
1081:
1080:
1059:
1058:
1037:
1036:
1015:
1010:
1009:
990:
989:
967:
889:
884:
883:
848:
843:
842:
823:
822:
803:
802:
780:
779:
751:
750:
704:polynomial ring
657:
656:
637:
636:
567:
566:
531:
530:
511:
510:
463:polynomial ring
441:
406:
373:
353:
352:
321:
313:
312:
291:
280:
279:
258:
250:
249:
227:
226:
223:
221:Degrees 0 and 1
160:
120:
117:
114:
111:
110:
88:
81:
61:
32:on Knowledge's
29:
12:
11:
5:
2415:
2413:
2405:
2404:
2399:
2389:
2388:
2385:
2384:
2383:
2382:
2334:
2331:
2312:explicitly say
2304:
2298:
2297:
2296:
2295:
2294:
2293:
2292:
2291:
2290:
2191:
2188:
2169:
2166:
2149:. The section
2119:
2113:
2112:
2111:
2110:
2109:
2108:
2107:
2106:
2105:
2104:
2103:
2102:
2101:
2037:can represent
1991:
1990:
1989:
1948:
1947:Degree of zero
1945:
1904:
1901:
1885:
1882:
1881:
1880:
1879:
1878:
1835:
1834:
1824:85.141.139.249
1795:
1792:
1777:
1774:
1773:
1772:
1754:
1751:
1748:
1745:
1742:
1739:
1736:
1716:
1713:
1710:
1707:
1702:
1698:
1694:
1691:
1686:
1682:
1678:
1675:
1670:
1666:
1662:
1659:
1656:
1636:
1610:
1607:
1606:
1605:
1604:
1603:
1602:
1601:
1600:
1599:
1598:
1597:
1596:
1595:
1558:
1554:
1542:
1539:
1536:
1533:
1530:
1527:
1522:
1518:
1513:
1509:
1504:
1501:
1498:
1495:
1492:
1487:
1483:
1479:
1476:
1473:
1470:
1450:
1447:
1444:
1441:
1438:
1435:
1430:
1426:
1421:
1417:
1412:
1409:
1406:
1403:
1400:
1397:
1394:
1391:
1388:
1385:
1380:
1376:
1371:
1367:
1362:
1359:
1356:
1336:
1333:
1330:
1327:
1324:
1321:
1318:
1314:
1310:
1305:
1302:
1299:
1279:
1274:
1271:
1268:
1265:
1262:
1240:
1228:
1225:
1222:
1219:
1214:
1210:
1188:
1167:
1146:
1140:
1136:
1132:
1129:
1126:
1123:
1101:
1097:
1091:
1088:
1079:, the mapping
1067:
1045:
1022:
1018:
997:
974:
970:
966:
963:
960:
957:
951:
946:
943:
940:
937:
934:
930:
925:
921:
918:
914:
910:
907:
904:
901:
896:
892:
871:
866:
863:
860:
855:
851:
830:
810:
799:
787:
767:
764:
761:
758:
747:
741:
740:
731:
707:
693:
681:
676:
673:
670:
667:
664:
644:
633:
632:
623:
611:
608:
605:
602:
599:
596:
593:
589:
584:
580:
577:
574:
554:
549:
545:
541:
538:
529:is a function
518:
501:
499:
497:
496:
487:
486:
485:
466:
440:
437:
436:
435:
405:
402:
401:
400:
380:
376:
372:
369:
366:
363:
360:
343:146.227.11.232
328:
324:
320:
298:
294:
290:
287:
265:
261:
257:
237:
234:
222:
219:
218:
217:
159:
156:
153:
152:
149:
148:
145:
144:
133:
127:
126:
124:
107:the discussion
94:
93:
77:
65:
64:
56:
44:
43:
37:
26:
13:
10:
9:
6:
4:
3:
2:
2414:
2403:
2400:
2398:
2395:
2394:
2392:
2381:
2377:
2373:
2369:
2368:
2367:
2363:
2359:
2355:
2354:
2353:
2352:
2348:
2344:
2340:
2332:
2330:
2329:
2325:
2321:
2317:
2313:
2309:
2303:
2299:
2289:
2286:
2282:
2281:
2280:
2276:
2272:
2268:
2264:
2260:
2256:
2252:
2251:
2250:
2246:
2242:
2238:
2234:
2230:
2226:
2225:
2224:
2221:
2216:
2215:
2214:
2213:
2209:
2205:
2186:
2178:
2174:
2167:
2165:
2164:
2160:
2156:
2152:
2148:
2144:
2140:
2136:
2132:
2128:
2124:
2118:
2114:
2100:
2097:
2093:
2092:
2091:
2087:
2083:
2079:
2075:
2074:
2073:
2069:
2065:
2061:
2057:
2056:
2055:
2052:
2032:
2031:
2030:
2026:
2022:
2018:
2014:
2010:
2006:
2005:
2004:
2000:
1996:
1992:
1987:
1984:
1983:
1981:
1980:
1979:
1978:
1974:
1970:
1966:
1962:
1958:
1953:
1946:
1944:
1943:
1939:
1935:
1930:
1928:
1924:
1920:
1919:66.183.55.146
1916:
1908:
1900:
1899:
1895:
1891:
1884:Hectic's link
1877:
1873:
1869:
1865:
1861:
1856:
1852:
1851:
1850:
1846:
1842:
1837:
1836:
1833:
1829:
1825:
1820:
1819:
1818:
1817:
1813:
1809:
1805:
1801:
1793:
1791:
1790:
1786:
1782:
1781:74.244.68.148
1775:
1771:
1768:
1752:
1749:
1743:
1737:
1734:
1714:
1711:
1708:
1705:
1700:
1696:
1692:
1689:
1684:
1680:
1676:
1673:
1668:
1664:
1660:
1657:
1654:
1634:
1626:
1625:
1624:
1623:
1619:
1608:
1594:
1591:
1587:
1586:
1585:
1582:
1578:
1577:
1576:
1573:
1569:
1568:
1567:
1564:
1537:
1534:
1528:
1520:
1516:
1507:
1502:
1499:
1493:
1485:
1481:
1474:
1471:
1468:
1445:
1442:
1436:
1428:
1424:
1415:
1410:
1407:
1401:
1395:
1392:
1386:
1378:
1374:
1365:
1360:
1357:
1331:
1328:
1322:
1316:
1308:
1303:
1300:
1277:
1272:
1266:
1260:
1251:
1250:
1249:
1246:
1241:
1226:
1223:
1220:
1217:
1212:
1208:
1165:
1130:
1127:
1124:
1121:
1099:
1095:
1086:
1020:
1016:
995:
972:
968:
961:
955:
949:
944:
938:
932:
919:
916:
912:
908:
902:
894:
890:
869:
861:
858:
853:
849:
828:
808:
800:
785:
762:
756:
748:
745:
744:
743:
742:
739:
736:
732:
728:
724:
723:
722:
721:
718:
714:
710:
705:
701:
697:
679:
674:
668:
662:
642:
630:
626:
625:
624:
606:
603:
597:
591:
578:
575:
552:
539:
536:
516:
509:
505:
495:
492:
488:
483:
479:
475:
471:
467:
464:
460:
456:
455:
452:
451:
450:
449:
446:
438:
434:
431:
427:
423:
419:
415:
411:
410:
409:
403:
399:
396:
378:
374:
370:
367:
364:
361:
358:
350:
349:
348:
347:
344:
326:
322:
318:
296:
292:
288:
285:
263:
259:
255:
235:
232:
220:
216:
212:
208:
203:
202:
201:
200:
196:
192:
187:
186:
183:
177:
173:
170:
166:
163:
157:
142:
138:
132:
129:
128:
125:
108:
104:
100:
99:
91:
85:
80:
78:
75:
71:
70:
66:
60:
57:
54:
50:
45:
41:
35:
27:
23:
18:
17:
2372:192.96.44.56
2343:192.96.44.56
2336:
2306:
2300:Degree is a
2271:50.53.53.229
2266:
2262:
2258:
2254:
2241:50.53.53.229
2233:propositions
2204:50.53.53.229
2171:
2155:50.53.53.229
2143:Term (logic)
2121:
2077:
2059:
1995:50.53.50.168
1950:
1931:
1913:— Preceding
1909:
1906:
1887:
1868:Double sharp
1803:
1799:
1797:
1779:
1612:
726:
712:
634:
498:
481:
477:
473:
469:
442:
425:
421:
417:
407:
224:
188:
178:
174:
171:
167:
164:
161:
137:Mid-priority
136:
96:
62:Mid‑priority
40:WikiProjects
2320:50.53.53.71
2021:50.53.55.20
1965:Spencerleet
1952:Spencerleet
1808:Georgia guy
1609:Degree of i
1035:because on
882:defined by
629:AdamSmithee
622:is finite.
395:AdamSmithee
191:Jewels Vern
182:AdamSmithee
112:Mathematics
103:mathematics
59:Mathematics
30:Start-class
2391:Categories
2253:Actually,
2064:TomS TDotO
696:polynomial
565:such that
504:polynomial
176:n P(m)=0.
2339:Trinomial
2310:does not
2173:This edit
2123:This edit
1617:pizza1512
430:Bo Jacoby
2358:D.Lazard
2308:The lead
2115:Link to
2082:D.Lazard
1969:D.Lazard
1934:D.Lazard
1915:unsigned
1767:Ceroklis
1590:Ceroklis
1572:Ceroklis
1553:and if P
1245:Ceroklis
717:Ceroklis
445:Ceroklis
207:D.Lazard
2285:Quondum
2220:Quondum
2177:section
2096:Quondum
2051:Quondum
1890:Sobeita
1841:Sobeita
1798:Why is
1581:grubber
1563:grubber
1461:, then
735:grubber
713:because
709:grubber
506:over a
491:grubber
139:on the
2235:, but
2231:lists
2139:merged
1804:centic
1800:hectic
36:scale.
2141:into
1727:. So
1200:with
1178:over
821:over
457:In a
2376:talk
2362:talk
2347:talk
2324:talk
2318:. --
2275:talk
2245:talk
2208:talk
2159:talk
2145:and
2137:was
2117:term
2086:talk
2068:talk
2025:talk
1999:talk
1973:talk
1938:talk
1923:talk
1894:talk
1872:talk
1855:here
1845:talk
1828:talk
1812:talk
1785:talk
1008:and
727:some
702:and
508:ring
211:talk
195:talk
2267:deg
2263:deg
2259:deg
2255:deg
2129:to
1806:??
1735:deg
1057:or
131:Mid
2393::
2378:)
2364:)
2349:)
2326:)
2277:)
2247:)
2210:)
2190:∞
2187:−
2161:)
2133:.
2088:)
2070:)
2047:−∞
2043:−1
2039:−∞
2035:−1
2027:)
2001:)
1993:--
1975:)
1959:,
1940:)
1925:)
1896:)
1874:)
1857::
1847:)
1830:)
1822:--
1814:)
1787:)
1738:
1535:≠
1503:∈
1494:∈
1443:≠
1411:∈
1402:⊆
1393:≠
1361:∈
1329:≠
1304:∈
1273:≠
1139:→
1090:↦
945:≠
920:∈
913:∑
865:→
698:,
675:≠
604:≠
579:∈
548:→
502:A
420:+5
319:10
233:10
213:)
197:)
2374:(
2360:(
2345:(
2322:(
2273:(
2243:(
2206:(
2157:(
2084:(
2078:x
2066:(
2060:x
2023:(
1997:(
1971:(
1936:(
1921:(
1892:(
1870:(
1843:(
1826:(
1810:(
1783:(
1753:0
1750:=
1747:)
1744:i
1741:(
1715:.
1712:.
1709:.
1706:+
1701:2
1697:z
1693:0
1690:+
1685:1
1681:z
1677:0
1674:+
1669:0
1665:z
1661:i
1658:=
1655:i
1635:i
1559:2
1555:1
1541:}
1538:0
1532:)
1529:n
1526:(
1521:2
1517:P
1512:|
1508:N
1500:n
1497:{
1491:)
1486:1
1482:P
1478:(
1475:g
1472:e
1469:d
1449:}
1446:0
1440:)
1437:n
1434:(
1429:2
1425:P
1420:|
1416:N
1408:n
1405:{
1399:}
1396:0
1390:)
1387:n
1384:(
1379:1
1375:P
1370:|
1366:N
1358:n
1355:{
1335:}
1332:0
1326:)
1323:n
1320:(
1317:P
1313:|
1309:N
1301:n
1298:{
1278:0
1270:)
1267:n
1264:(
1261:P
1239:.
1227:p
1224:x
1221:e
1218:=
1213:P
1209:f
1187:R
1166:P
1145:R
1135:R
1131::
1128:p
1125:x
1122:e
1100:P
1096:f
1087:P
1066:C
1044:R
1021:P
1017:f
996:P
973:n
969:x
965:)
962:n
959:(
956:P
950:0
942:)
939:n
936:(
933:P
929:|
924:N
917:n
909:=
906:)
903:x
900:(
895:P
891:f
870:R
862:R
859::
854:P
850:f
829:R
809:P
798:.
786:X
766:)
763:n
760:(
757:P
680:0
672:)
669:n
666:(
663:P
643:P
610:}
607:0
601:)
598:n
595:(
592:P
588:|
583:N
576:n
573:{
553:R
544:N
540::
537:P
517:R
482:b
480:+
478:a
474:b
470:a
426:x
422:x
418:x
379:0
375:X
371:5
368:=
365:5
362:=
359:f
327:2
323:x
297:1
293:5
289:=
286:5
264:0
260:x
256:5
236:x
209:(
193:(
143:.
42::
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
