Consider the
If we choose to set all
If we go higher,
call this bilinear identity for , but there are F_{16}non-bilinear ones which is allowed by Pfister’s theorem (1965). (Author’s note: The situation seems reminiscent of the unqualified statement that there is no formula for the general quintic. No formula just in the radicals, but if we expand our tool-set of functions, there is one using elliptic functions as shown by Hermite.)
There are various formulations but, in this author’s version, it will be given in _{i}.)
After Euler’s discovery of the four-square, and the eight-square by Degen, Graves, and Cayley, it was reasonable to search for a sixteen-square version. However, given an identity of form,
(x
Hurwitz’s theorem (1898) states that the only possible identities of this sort are for n = {1, 2, 4, 8}, hence there are only four normed division algebras over the reals: the real numbers, complex numbers, quaternions, and octonions.
But there are ways we can evade Hurwitz’s theorem. First, we can drop the requirement that
this guarantees a bilinear identity for
(x
z z z z
and so on.
The second way is to retain
Note also the incidental fact that,
Similarly to the previous,
Not surprisingly,
With all the variables
The 16-square (and analogously for the two others) has several interesting properties:
which is one of the simplest possible conditions for a “quasi-bilinear” 16-square identity (at the cost of one variable being linearly dependent on the others).
u_{i} vanishes, as well as the second powers in the numerators, and we have a fully bilinear form. If any seven of the y_{i} are also set to zero, then we get the nice form,
or a bilinear [9.9.16] which, for
The author wishes to thank Keith Conrad for his paper,
For those who wish to verify the 16-square, the LaTex code for the
<math>\begin{align} &^{z1 \,=\, x1 y1 - x2 y2 - x3 y3 - x4 y4 - x5 y5 - x6 y6 - x7 y7 - x8 y8 + u1 y9 - u2 y10 - u3 y11 - u4 y12 - u5 y13 - u6 y14 - u7 y15 - u8 y16}\\ &^{z2 \,=\, x2 y1 + x1 y2 + x4 y3 - x3 y4 + x6 y5 - x5 y6 - x8 y7 + x7 y8 + u2 y9 + u1 y10 + u4 y11 - u3 y12 + u6 y13 - u5 y14 - u8 y15 + u7 y16}\\ &^{z3 \,=\, x3 y1 - x4 y2 + x1 y3 + x2 y4 + x7 y5 + x8 y6 - x5 y7 - x6 y8 + u3 y9 - u4 y10 + u1 y11 + u2 y12 + u7 y13 + u8 y14 - u5 y15 - u6 y16}\\ &^{z4 \,=\, x4 y1 + x3 y2 - x2 y3 + x1 y4 + x8 y5 - x7 y6 + x6 y7 - x5 y8 + u4 y9 + u3 y10 - u2 y11 + u1 y12 + u8 y13 - u7 y14 + u6 y15 - u5 y16}\\ &^{z5 \,=\, x5 y1 - x6 y2 - x7 y3 - x8 y4 + x1 y5 + x2 y6 + x3 y7 + x4 y8 + u5 y9 - u6 y10 - u7 y11 - u8 y12 + u1 y13 + u2 y14 + u3 y15 + u4 y16}\\ &^{z6 \,=\, x6 y1 + x5 y2 - x8 y3 + x7 y4 - x2 y5 + x1 y6 - x4 y7 + x3 y8 + u6 y9 + u5 y10 - u8 y11 + u7 y12 - u2 y13 + u1 y14 - u4 y15 + u3 y16}\\ &^{z7 \,=\, x7 y1 + x8 y2 + x5 y3 - x6 y4 - x3 y5 + x4 y6 + x1 y7 - x2 y8 + u7 y9 + u8 y10 + u5 y11 - u6 y12 - u3 y13 + u4 y14 + u1 y15 - u2 y16}\\ &^{z8 \,=\, x8 y1 - x7 y2 + x6 y3 + x5 y4 - x4 y5 - x3 y6 + x2 y7 + x1 y8 + u8 y9 - u7 y10 + u6 y11 + u5 y12 - u4 y13 - u3 y14 + u2 y15 + u1 y16}\\ &^{z9 \,=\, x9 y1 - x10 y2 - x11 y3 - x12 y4 - x13 y5 - x14 y6 - x15 y7 - x16 y8 + x1 y9 - x2 y10 - x3 y11 - x4 y12 - x5 y13 - x6 y14 - x7 y15 - x8 y16}\\ &^{z10 \,=\, x10 y1 + x9 y2 + x12 y3 - x11 y4 + x14 y5 - x13 y6 - x16 y7 + x15 y8 + x2 y9 + x1 y10 + x4 y11 - x3 y12 + x6 y13 - x5 y14 - x8 y15 + x7 y16}\\ &^{z11 \,=\, x11 y1 - x12 y2 + x9 y3 + x10 y4 + x15 y5 + x16 y6 - x13 y7 - x14 y8 + x3 y9 - x4 y10 + x1 y11 + x2 y12 + x7 y13 + x8 y14 - x5 y15 - x6 y16}\\ &^{z12 \,=\, x12 y1 + x11 y2 - x10 y3 + x9 y4 + x16 y5 - x15 y6 + x14 y7 - x13 y8 + x4 y9 + x3 y10 - x2 y11 + x1 y12 + x8 y13 - x7 y14 + x6 y15 - x5 y16}\\ &^{z13 \,=\, x13 y1 - x14 y2 - x15 y3 - x16 y4 + x9 y5 + x10 y6 + x11 y7 + x12 y8 + x5 y9 - x6 y10 - x7 y11 - x8 y12 + x1 y13 + x2 y14 + x3 y15 + x4 y16}\\ &^{z14 \,=\, x14 y1 + x13 y2 - x16 y3 + x15 y4 - x10 y5 + x9 y6 - x12 y7 + x11 y8 + x6 y9 + x5 y10 - x8 y11 + x7 y12 - x2 y13 + x1 y14 - x4 y15 + x3 y16}\\ &^{z15 \,=\, x15 y1 + x16 y2 + x13 y3 - x14 y4 - x11 y5 + x12 y6 + x9 y7 - x10 y8 + x7 y9 + x8 y10 + x5 y11 - x6 y12 - x3 y13 + x4 y14 + x1 y15 - x2 y16}\\ &^{z16 \,=\, x16 y1 - x15 y2 + x14 y3 + x13 y4 - x12 y5 - x11 y6 + x10 y7 + x9 y8 + x8 y9 - x7 y10 + x6 y11 + x5 y12 - x4 y13 - x3 y14 + x2 y15 + x1 y16}\\ \end{align}</math>
<math>\begin{align} &^{u1 \,=\,} \tfrac{(-x1^2+x2^2+x3^2+x4^2+x5^2+x6^2+x7^2+x8^2)x9 - 2x1(0 x1 x9 +x2 x10 +x3 x11 +x4 x12 +x5 x13 +x6 x14 +x7 x15 +x8 x16)}{d}\\ &^{u2 \,=\,} \tfrac{(x1^2-x2^2+x3^2+x4^2+x5^2+x6^2+x7^2+x8^2)x10 - 2x2(x1 x9 + 0 x2 x10 +x3 x11 +x4 x12 +x5 x13 +x6 x14 +x7 x15 +x8 x16)}{d}\\ &^{u3 \,=\,} \tfrac{(x1^2+x2^2-x3^2+x4^2+x5^2+x6^2+x7^2+x8^2)x11 - 2x3(x1 x9 +x2 x10 + 0 x3 x11 +x4 x12 +x5 x13 +x6 x14 +x7 x15 +x8 x16)}{d}\\ \vdots\\ &^{u8 \,=\,} \tfrac{(x1^2+x2^2+x3^2+x4^2+x5^2+x6^2+x7^2-x8^2)x16 - 2x8(x1 x9 +x2 x10 +x3 x11 +x4 x12 +x5 x13 +x6 x14 +x7 x15 + 0 x8 x16)}{d}\\ \,&^{d \,=\, x1^2+x2^2+x3^2+x4^2+x5^2+x6^2+x7^2+x8^2} \end{align}</math>
© Tito Piezas III, Jan 2012 (Modified Feb 2012) You can email author at
tpiezas@gmail.com. |