Pfister's sixteen-square identity

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In algebra, Pfister's sixteen-square identity is a non-bilinear identity of form [math]\displaystyle{ \left(x_1^2+x_2^2+x_3^2+\cdots+x_{16}^2\right)\left(y_1^2+y_2^2+y_3^2+\cdots+y_{16}^2\right) = z_1^2+z_2^2+z_3^2+\cdots+z_{16}^2 }[/math]

It was first proven to exist by H. Zassenhaus and W. Eichhorn in the 1960s,[1] and independently by Albrecht Pfister[2] around the same time. There are several versions, a concise one of which is

[math]\displaystyle{ \begin{align} &\scriptstyle{z_1 = {\color{blue}{x_1 y_1 - x_2 y_2 - x_3 y_3 - x_4 y_4 - x_5 y_5 - x_6 y_6 - x_7 y_7 - x_8 y_8}} + u_1 y_9 - u_2 y_{10} - u_3 y_{11} - u_4 y_{12} - u_5 y_{13} - u_6 y_{14} - u_7 y_{15} - u_8 y_{16}} \\ &\scriptstyle{z_2 = {\color{blue}{x_2 y_1 + x_1 y_2 + x_4 y_3 - x_3 y_4 + x_6 y_5 - x_5 y_6 - x_8 y_7 + x_7 y_8}} + u_2 y_9 + u_1 y_{10} + u_4 y_{11} - u_3 y_{12} + u_6 y_{13} - u_5 y_{14} - u_8 y_{15} + u_7 y_{16}} \\ &\scriptstyle{z_3 = {\color{blue}{x_3 y_1 - x_4 y_2 + x_1 y_3 + x_2 y_4 + x_7 y_5 + x_8 y_6 - x_5 y_7 - x_6 y_8}} + u_3 y_9 - u_4 y_{10} + u_1 y_{11} + u_2 y_{12} + u_7 y_{13} + u_8 y_{14} - u_5 y_{15} - u_6 y_{16}} \\ &\scriptstyle{z_4 = {\color{blue}{x_4 y_1 + x_3 y_2 - x_2 y_3 + x_1 y_4 + x_8 y_5 - x_7 y_6 + x_6 y_7 - x_5 y_8}} + u_4 y_9 + u_3 y_{10} - u_2 y_{11} + u_1 y_{12} + u_8 y_{13} - u_7 y_{14} + u_6 y_{15} - u_5 y_{16}} \\ &\scriptstyle{z_5 = {\color{blue}{x_5 y_1 - x_6 y_2 - x_7 y_3 - x_8 y_4 + x_1 y_5 + x_2 y_6 + x_3 y_7 + x_4 y_8}} + u_5 y_9 - u_6 y_{10} - u_7 y_{11} - u_8 y_{12} + u_1 y_{13} + u_2 y_{14} + u_3 y_{15} + u_4 y_{16}} \\ &\scriptstyle{z_6 = {\color{blue}{x_6 y_1 + x_5 y_2 - x_8 y_3 + x_7 y_4 - x_2 y_5 + x_1 y_6 - x_4 y_7 + x_3 y_8}} + u_6 y_9 + u_5 y_{10} - u_8 y_{11} + u_7 y_{12} - u_2 y_{13} + u_1 y_{14} - u_4 y_{15} + u_3 y_{16}} \\ &\scriptstyle{z_7 = {\color{blue}{x_7 y_1 + x_8 y_2 + x_5 y_3 - x_6 y_4 - x_3 y_5 + x_4 y_6 + x_1 y_7 - x_2 y_8}} + u_7 y_9 + u_8 y_{10} + u_5 y_{11} - u_6 y_{12} - u_3 y_{13} + u_4 y_{14} + u_1 y_{15} - u_2 y_{16}} \\ &\scriptstyle{z_8 = {\color{blue}{x_8 y_1 - x_7 y_2 + x_6 y_3 + x_5 y_4 - x_4 y_5 - x_3 y_6 + x_2 y_7 + x_1 y_8}} + u_8 y_9 - u_7 y_{10} + u_6 y_{11} + u_5 y_{12} - u_4 y_{13} - u_3 y_{14} + u_2 y_{15} + u_1 y_{16}} \\ &\scriptstyle{z_9 = x_9 y_1 - x_{10} y_2 - x_{11} y_3 - x_{12} y_4 - x_{13} y_5 - x_{14} y_6 - x_{15} y_7 - x_{16} y_8 + x_1 y_9 - x_2 y_{10} - x_3 y_{11} - x_4 y_{12} - x_5 y_{13} - x_6 y_{14} - x_7 y_{15} - x_8 y_{16}} \\ &\scriptstyle{z_{10} = x_{10} y_1 + x_9 y_2 + x_{12} y_3 - x_{11} y_4 + x_{14} y_5 - x_{13} y_6 - x_{16} y_7 + x_{15} y_8 + x_2 y_9 + x_1 y_{10} + x_4 y_{11} - x_3 y_{12} + x_6 y_{13} - x_5 y_{14} - x_8 y_{15} + x_7 y_{16}} \\ &\scriptstyle{z_{11} = x_{11} y_1 - x_{12} y_2 + x_9 y_3 + x_{10} y_4 + x_{15} y_5 + x_{16} y_6 - x_{13} y_7 - x_{14} y_8 + x_3 y_9 - x_4 y_{10} + x_1 y_{11} + x_2 y_{12} + x_7 y_{13} + x_8 y_{14} - x_5 y_{15} - x_6 y_{16}} \\ &\scriptstyle{z_{12} = x_{12} y_1 + x_{11} y_2 - x_{10} y_3 + x_9 y_4 + x_{16} y_5 - x_{15} y_6 + x_{14} y_7 - x_{13} y_8 + x_4 y_9 + x_3 y_{10} - x_2 y_{11} + x_1 y_{12} + x_8 y_{13} - x_7 y_{14} + x_6 y_{15} - x_5 y_{16}} \\ &\scriptstyle{z_{13} = x_{13} y_1 - x_{14} y_2 - x_{15} y_3 - x_{16} y_4 + x_9 y_5 + x_{10} y_6 + x_{11} y_7 + x_{12} y_8 + x_5 y_9 - x_6 y_{10} - x_7 y_{11} - x_8 y_{12} + x_1 y_{13} + x_2 y_{14} + x_3 y_{15} + x_4 y_{16}} \\ &\scriptstyle{z_{14} = x_{14} y_1 + x_{13} y_2 - x_{16} y_3 + x_{15} y_4 - x_{10} y_5 + x_9 y_6 - x_{12} y_7 + x_{11} y_8 + x_6 y_9 + x_5 y_{10} - x_8 y_{11} + x_7 y_{12} - x_2 y_{13} + x_1 y_{14} - x_4 y_{15} + x_3 y_{16}} \\ &\scriptstyle{z_{15} = x_{15} y_1 + x_{16} y_2 + x_{13} y_3 - x_{14} y_4 - x_{11} y_5 + x_{12} y_6 + x_9 y_7 - x_{10} y_8 + x_7 y_9 + x_8 y_{10} + x_5 y_{11} - x_6 y_{12} - x_3 y_{13} + x_4 y_{14} + x_1 y_{15} - x_2 y_{16}} \\ &\scriptstyle{z_{16} = x_{16} y_1 - x_{15} y_2 + x_{14} y_3 + x_{13} y_4 - x_{12} y_5 - x_{11} y_6 + x_{10} y_7 + x_9 y_8 + x_8 y_9 - x_7 y_{10} + x_6 y_{11} + x_5 y_{12} - x_4 y_{13} - x_3 y_{14} + x_2 y_{15} + x_1 y_{16}} \end{align} }[/math]

If all [math]\displaystyle{ x_i }[/math] and [math]\displaystyle{ y_i }[/math] with [math]\displaystyle{ i\gt 8 }[/math] are set equal to zero, then it reduces to Degen's eight-square identity (in blue). The [math]\displaystyle{ u_i }[/math] are

[math]\displaystyle{ \begin{align} &u_1 = \tfrac{\left(ax_1^2+x_2^2+x_3^2+x_4^2+x_5^2+x_6^2+x_7^2+x_8^2\right)x_9 - 2x_1\left(bx_1 x_9 +x_2 x_{10} +x_3 x_{11} +x_4 x_{12} +x_5 x_{13} +x_6 x_{14} +x_7 x_{15} +x_8 x_{16}\right)}{c} \\ &u_2 = \tfrac{\left(x_1^2+ax_2^2+x_3^2+x_4^2+x_5^2+x_6^2+x_7^2+x_8^2\right)x_{10} - 2x_2\left(x_1 x_9 +bx_2 x_{10} +x_3 x_{11} +x_4 x_{12} +x_5 x_{13} +x_6 x_{14} +x_7 x_{15} +x_8 x_{16}\right)}{c} \\ &u_3 = \tfrac{\left(x_1^2+x_2^2+ax_3^2+x_4^2+x_5^2+x_6^2+x_7^2+x_8^2\right)x_{11} - 2x_3\left(x_1 x_9 +x_2 x_{10} +bx_3 x_{11} +x_4 x_{12} +x_5 x_{13} +x_6 x_{14} +x_7 x_{15} +x_8 x_{16}\right)}{c} \\ &u_4 = \tfrac{\left(x_1^2+x_2^2+x_3^2+ax_4^2+x_5^2+x_6^2+x_7^2+x_8^2\right)x_{12} - 2x_4\left(x_1 x_9 +x_2 x_{10} +x_3 x_{11} +bx_4 x_{12} +x_5 x_{13} +x_6 x_{14} +x_7 x_{15} +x_8 x_{16}\right)}{c} \\ &u_5 = \tfrac{\left(x_1^2+x_2^2+x_3^2+x_4^2+ax_5^2+x_6^2+x_7^2+x_8^2\right)x_{13} - 2x_5\left(x_1 x_9 +x_2 x_{10} +x_3 x_{11} +x_4 x_{12} +bx_5 x_{13} +x_6 x_{14} +x_7 x_{15} +x_8 x_{16}\right)}{c} \\ &u_6 = \tfrac{\left(x_1^2+x_2^2+x_3^2+x_4^2+x_5^2+ax_6^2+x_7^2+x_8^2\right)x_{14} - 2x_6\left(x_1 x_9 +x_2 x_{10} +x_3 x_{11} +x_4 x_{12} +x_5 x_{13} +bx_6 x_{14} +x_7 x_{15} +x_8 x_{16}\right)}{c} \\ &u_7 = \tfrac{\left(x_1^2+x_2^2+x_3^2+x_4^2+x_5^2+x_6^2+ax_7^2+x_8^2\right)x_{15} - 2x_7\left(x_1 x_9 +x_2 x_{10} +x_3 x_{11} +x_4 x_{12} +x_5 x_{13} +x_6 x_{14} +bx_7 x_{15} +x_8 x_{16}\right)}{c} \\ &u_8 = \tfrac{\left(x_1^2+x_2^2+x_3^2+x_4^2+x_5^2+x_6^2+x_7^2+ax_8^2\right)x_{16} - 2x_8\left(x_1 x_9 +x_2 x_{10} +x_3 x_{11} +x_4 x_{12} +x_5 x_{13} +x_6 x_{14} +x_7 x_{15} +bx_8 x_{16}\right)}{c} \end{align} }[/math]

and,

[math]\displaystyle{ a=-1,\;\;b=0,\;\;c=x_1^2+x_2^2+x_3^2+x_4^2+x_5^2+x_6^2+x_7^2+x_8^2\,. }[/math]

The identity shows that, in general, the product of two sums of sixteen squares is the sum of sixteen rational squares. Incidentally, the [math]\displaystyle{ u_i }[/math] also obey,

[math]\displaystyle{ u_1^2+u_2^2+u_3^2+u_4^2+u_5^2+u_6^2+u_7^2+u_8^2 = x_{9}^2+x_{10}^2+x_{11}^2+x_{12}^2+x_{13}^2+x_{14}^2+x_{15}^2+x_{16}^2 }[/math]

No sixteen-square identity exists involving only bilinear functions since Hurwitz's theorem states an identity of the form

[math]\displaystyle{ \left(x_1^2+x_2^2+x_3^2+\cdots+x_n^2)(y_1^2+y_2^2+y_3^2+\cdots+y_n^2\right) = z_1^2+z_2^2+z_3^2+\cdots+z_n^2 }[/math]

with the [math]\displaystyle{ z_i }[/math] bilinear functions of the [math]\displaystyle{ x_i }[/math] and [math]\displaystyle{ y_i }[/math] is possible only for n ∈ {1, 2, 4, 8} . However, the more general Pfister's theorem (1965) shows that if the [math]\displaystyle{ z_i }[/math] are rational functions of one set of variables, hence has a denominator, then it is possible for all [math]\displaystyle{ n = 2^m }[/math].[3] There are also non-bilinear versions of Euler's four-square and Degen's eight-square identities.

See also

References

  1. ↑ H. Zassenhaus and W. Eichhorn, "Herleitung von Acht- und Sechzehn-Quadrate-Identitäten mit Hilfe von Eigenschaften der verallgemeinerten Quaternionen und der Cayley-Dicksonchen Zahlen," Arch. Math. 17 (1966), 492-496
  2. ↑ A. Pfister, Zur Darstellung von -1 als Summe von Quadraten in einem Körper," J. London Math. Soc. 40 (1965), 159-165
  3. ↑ Pfister's Theorem on Sums of Squares, Keith Conrad, http://www.math.uconn.edu/~kconrad/blurbs/linmultialg/pfister.pdf

External links



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