Timeline for Surprising behaviour of polynomial that generates the series 1,2,4,8,...2^(k-1)
Current License: CC BY-SA 2.5
3 events
| when toggle format | what | by | license | comment | |
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| Mar 28, 2011 at 5:54 | comment | added | Michael | More generally, for $(a_1,\ldots,a_n) \in F^n$, the polynomial $p_{(a_1,\ldots,a_n)} = \prod_{i=1}^n \left(\prod_{a \neq a_i} \frac{X_i - a}{a_i - a} \right)$ in $F[X_1,\ldots,X_n]$ evaluates to $1$ or $0$ according as $(X_1,\ldots,X_n)$ is put equal to $(a_1,\ldots,a_n)$ or otherwise. Taking linear combinations, it follows that every function $F^n \to F$ arises from polynomial in $n$ variables. | |
| Mar 28, 2011 at 5:52 | comment | added | Michael | I like this construction because it demonstrates that, for a finite field $F$, all of the functions $F \to F$ are polynomial functions. | |
| Oct 16, 2010 at 18:58 | history | answered | Alfonso Gracia-Saz | CC BY-SA 2.5 |