521:
343:
232:
395:
261:
378:
619:
614:
609:
164:
242:
358:
108:
96:
44:
516:{\displaystyle c_{1}x_{1}^{r}+\cdots +c_{n}x_{n}^{r}=0,\quad c_{i}\in \mathbb {Z} ,\ i=1,\ldots ,n}
354:
586:
572:
28:
245:
377:. Essential to the proof is a special case, which can be proved by an application of the
553:
60:
603:
577:
100:
20:
590:
575:(1957). "Homogeneous forms of odd degree in a large number of variables".
31:, is a statement about the representability of zero by odd degree forms.
527:
338:{\displaystyle f_{1}(x)=\cdots =f_{k}(x)=0{\text{ for all }}x\in V.}
16:
Statement about the representability of zero by odd degree forms
398:
264:
167:
227:{\displaystyle n\geq \psi (r_{1},\ldots ,r_{k},l,K)}
515:
337:
226:
552:is necessary, since even degree forms, such as
8:
556:, may take the value 0 only at the origin.
482:
481:
472:
452:
447:
437:
418:
413:
403:
397:
318:
297:
269:
263:
203:
184:
166:
564:
381:, of the theorem which states that if
130:variables. Then there exists a number
361:over the maximal degree of the forms
7:
14:
554:positive definite quadratic forms
545:, not all of which are 0.
467:
379:Hardy–Littlewood circle method
309:
303:
281:
275:
221:
177:
1:
79:be odd natural numbers, and
35:Statement of Birch's theorem
636:
389:is odd, then the equation
385:is sufficiently large and
620:Theorems in number theory
591:10.1112/S0025579300001145
548:The restriction to odd
97:homogeneous polynomials
34:
615:Analytic number theory
517:
339:
228:
45:algebraic number field
610:Diophantine equations
518:
357:of the theorem is by
340:
237:then there exists an
229:
396:
262:
165:
457:
423:
320: for all
526:has a solution in
513:
443:
409:
335:
224:
536:, ...,
491:
368:, ...,
321:
141:, ...,
117:, ...,
86:, ...,
70:, ...,
627:
595:
594:
569:
522:
520:
519:
514:
489:
485:
477:
476:
456:
451:
442:
441:
422:
417:
408:
407:
344:
342:
341:
336:
322:
319:
302:
301:
274:
273:
233:
231:
230:
225:
208:
207:
189:
188:
126:respectively in
29:Bryan John Birch
635:
634:
630:
629:
628:
626:
625:
624:
600:
599:
598:
571:
570:
566:
562:
544:
535:
468:
433:
399:
394:
393:
376:
367:
351:
293:
265:
260:
259:
246:vector subspace
199:
180:
163:
162:
158:) such that if
149:
140:
125:
116:
94:
85:
78:
69:
61:natural numbers
37:
25:Birch's theorem
17:
12:
11:
5:
633:
631:
623:
622:
617:
612:
602:
601:
597:
596:
563:
561:
558:
540:
533:
524:
523:
512:
509:
506:
503:
500:
497:
494:
488:
484:
480:
475:
471:
466:
463:
460:
455:
450:
446:
440:
436:
432:
429:
426:
421:
416:
412:
406:
402:
372:
365:
350:
347:
346:
345:
334:
331:
328:
325:
317:
314:
311:
308:
305:
300:
296:
292:
289:
286:
283:
280:
277:
272:
268:
235:
234:
223:
220:
217:
214:
211:
206:
202:
198:
195:
192:
187:
183:
179:
176:
173:
170:
145:
138:
121:
114:
90:
83:
74:
67:
36:
33:
15:
13:
10:
9:
6:
4:
3:
2:
632:
621:
618:
616:
613:
611:
608:
607:
605:
592:
588:
584:
580:
579:
574:
568:
565:
559:
557:
555:
551:
546:
543:
539:
532:
529:
510:
507:
504:
501:
498:
495:
492:
486:
478:
473:
469:
464:
461:
458:
453:
448:
444:
438:
434:
430:
427:
424:
419:
414:
410:
404:
400:
392:
391:
390:
388:
384:
380:
375:
371:
364:
360:
356:
348:
332:
329:
326:
323:
315:
312:
306:
298:
294:
290:
287:
284:
278:
270:
266:
258:
257:
256:
254:
250:
247:
244:
240:
218:
215:
212:
209:
204:
200:
196:
193:
190:
185:
181:
174:
171:
168:
161:
160:
159:
157:
153:
148:
144:
137:
133:
129:
124:
120:
113:
110:
106:
102:
98:
93:
89:
82:
77:
73:
66:
62:
58:
54:
50:
46:
42:
32:
30:
26:
22:
582:
576:
573:Birch, B. J.
567:
549:
547:
541:
537:
530:
525:
386:
382:
373:
369:
362:
352:
252:
248:
238:
236:
155:
151:
146:
142:
135:
131:
127:
122:
118:
111:
104:
101:coefficients
91:
87:
80:
75:
71:
64:
56:
52:
48:
40:
38:
27:, named for
24:
18:
585:: 102–105.
578:Mathematika
243:dimensional
21:mathematics
604:Categories
560:References
255:such that
505:…
479:∈
428:⋯
359:induction
327:∈
288:⋯
194:…
175:ψ
172:≥
528:integers
349:Remarks
154:,
150:,
109:degrees
490:
43:be an
355:proof
99:with
353:The
55:and
39:Let
587:doi
251:of
107:of
103:in
95:be
59:be
19:In
606::
581:.
63:,
51:,
47:,
23:,
593:.
589::
583:4
550:r
542:n
538:x
534:1
531:x
511:n
508:,
502:,
499:1
496:=
493:i
487:,
483:Z
474:i
470:c
465:,
462:0
459:=
454:r
449:n
445:x
439:n
435:c
431:+
425:+
420:r
415:1
411:x
405:1
401:c
387:r
383:n
374:k
370:f
366:1
363:f
333:.
330:V
324:x
316:0
313:=
310:)
307:x
304:(
299:k
295:f
291:=
285:=
282:)
279:x
276:(
271:1
267:f
253:K
249:V
241:-
239:l
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219:K
216:,
213:l
210:,
205:k
201:r
197:,
191:,
186:1
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178:(
169:n
156:K
152:l
147:k
143:r
139:1
136:r
134:(
132:ψ
128:n
123:k
119:r
115:1
112:r
105:K
92:k
88:f
84:1
81:f
76:k
72:r
68:1
65:r
57:n
53:l
49:k
41:K
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