Firoozbakht’s conjecture: Difference between revisions
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In mathematics, the '''Hurwitz problem''', named after [[Adolf Hurwitz]], is the problem of finding multiplicative relations between [[quadratic form]]s which generalise those known to exists between sums of squares in certain numbers of variables. | |||
There are well-known multiplicative relationships between sums of squares in two variables | |||
:<math> (x^2+y^2)(u^2+v^2) = (xu-yv)^2 + (xv+yu)^2 \ , </math> | |||
(known as the [[Brahmagupta–Fibonacci identity]]), and also [[Euler's four-square identity]] and [[Degen's eight-square identity]]. These may be interpreted as muliplicativity for the norms on the [[complex number]]s, [[quaternion]]s and [[octonion]]s respectively.<ref>Rajwade (1993) pp. 1–3</ref> | |||
The Hurwitz problem for the field ''K'' is to find general relations of the form | |||
:<math> (x_1^2+\cdots+x_r^2) \cdot (y_1^2+\cdots+y_s^2) = (z_1^2 + \cdots + z_n^2) \ , </math> | |||
with the ''z'' being bilinear forms in the ''x'' and ''y'': that is, each ''z'' is a ''K''-linear combination of terms of the form ''x''<sub>''i''</sub>''y''<sub>''j''</sub>.<ref name=Lam127>Lam (2005) p. 127</ref> We call a triple (''r'', ''s'', ''n'') ''admissible'' for ''K'' if such an identity exists.<ref name=R125>Rajwade (1993) p. 125</ref> Trivial cases of admissible triples include (''r'', ''s'', ''rs''). The problem is uninteresting for ''K'' of characteristic 2, since over such fields every sum of squares is a square, and we exclude this case. It is believed that otherwise admissibility is independent of the field of definition.<ref name=R137>Rajwade (1993) p. 137</ref> | |||
Hurwitz posed the problem in 1898 in the special case ''r'' = ''s'' = ''n'' and showed that, when coefficients are taken in '''C''', the only admissible values (''n'', ''n'', ''n'') were ''n'' = 1, 2, 4, 8:<ref name=Lam130>Lam (2005) p. 130</ref> his proof extends to any field of characteristic not 2.<ref name=R3>Rajwade (1993) p. 3</ref> | |||
The "Hurwitz–Radon" problem is that of finding admissible triples of the form (''r'', ''n'', ''n''). Obviously (1, ''n'', ''n'') is admissible. The '''Hurwitz–Radon theorem''' states that (ρ(''n''), ''n'', ''n'') is admissible over any field where ρ(''n'') is the function defined for ''n'' = 2<sup>''u''</sup>''v'', ''v'' odd, ''u'' = 4''a'' + ''b'', 0 ≤ ''b'' ≤ 3, as ''ρ''(''n'') = 8''a'' + 2<sup>''b''</sup>.<ref name=Lam130/><ref name=R137>Rajwade (1993) p. 137</ref> | |||
Other admissible triples include (3,5,7)<ref name=R138>Rajwade (1993) p. 138</ref> and (10, 10, 16).<ref name=R137/> | |||
==See also== | |||
* [[Composition algebra]] | |||
* [[Hurwitz's theorem (normed division algebras)]] | |||
* [[Radon–Hurwitz number]] | |||
==References== | |||
{{reflist}} | |||
* {{cite book | title=Introduction to Quadratic Forms over Fields | volume=67 | series=Graduate Studies in Mathematics | first=Tsit-Yuen | last=Lam | author-link=Tsit Yuen Lam | publisher=[[American Mathematical Society]] | year=2005 | isbn=0-8218-1095-2 | zbl=1068.11023 | mr = 2104929 }} | |||
* {{cite book | title=Squares | volume=171 | series=London Mathematical Society Lecture Note Series | first=A. R. | last=Rajwade | publisher=[[Cambridge University Press]] | year=1993 | isbn=0-521-42668-5 | zbl=0785.11022 }} | |||
{{algebra-stub}} | |||
[[Category:Field theory]] | |||
[[Category:Quadratic forms]] | |||
[[Category:Mathematical problems]] | |||
Latest revision as of 20:46, 25 December 2013
In mathematics, the Hurwitz problem, named after Adolf Hurwitz, is the problem of finding multiplicative relations between quadratic forms which generalise those known to exists between sums of squares in certain numbers of variables.
There are well-known multiplicative relationships between sums of squares in two variables
(known as the Brahmagupta–Fibonacci identity), and also Euler's four-square identity and Degen's eight-square identity. These may be interpreted as muliplicativity for the norms on the complex numbers, quaternions and octonions respectively.[1]
The Hurwitz problem for the field K is to find general relations of the form
with the z being bilinear forms in the x and y: that is, each z is a K-linear combination of terms of the form xiyj.[2] We call a triple (r, s, n) admissible for K if such an identity exists.[3] Trivial cases of admissible triples include (r, s, rs). The problem is uninteresting for K of characteristic 2, since over such fields every sum of squares is a square, and we exclude this case. It is believed that otherwise admissibility is independent of the field of definition.[4]
Hurwitz posed the problem in 1898 in the special case r = s = n and showed that, when coefficients are taken in C, the only admissible values (n, n, n) were n = 1, 2, 4, 8:[5] his proof extends to any field of characteristic not 2.[6]
The "Hurwitz–Radon" problem is that of finding admissible triples of the form (r, n, n). Obviously (1, n, n) is admissible. The Hurwitz–Radon theorem states that (ρ(n), n, n) is admissible over any field where ρ(n) is the function defined for n = 2uv, v odd, u = 4a + b, 0 ≤ b ≤ 3, as ρ(n) = 8a + 2b.[5][4]
Other admissible triples include (3,5,7)[7] and (10, 10, 16).[4]
See also
References
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