Timeline for Are the inverses of a set of quadratic polynomials linearly independent?
Current License: CC BY-SA 3.0
12 events
| when toggle format | what | by | license | comment | |
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| Jan 2, 2018 at 22:50 | comment | added | Noam D. Elkies | You can't edit a comment once 5 minutes have passed, but you can still delete and replace your own comment. | |
| Jan 2, 2018 at 21:37 | answer | added | Aaron Meyerowitz | timeline score: 1 | |
| Jan 2, 2018 at 20:44 | comment | added | Aaron Meyerowitz | Sorry, can't delete or edit earlier comment.It is true that $\sum_{a=0}^{p-1}\frac{1}{(x+a)(x+a+1)}=0$ as a formal polynomial. However there is no $x$ in $\mathbb{Z}_p$ for which it is defined. That might not matter for the question. At any rate, it seems legitimate for finite fields of order $p^e \gt p.$ | |
| Jan 2, 2018 at 20:41 | comment | added | Aaron Meyerowitz | It is true that $sum\limits_{a=0}^{p-1}\dfrac{1}{(x+a)(x+a+1)}=0\$ as a formal polynomial. However there is no $x$ in $\mathb{Z}_p$ for which it is defined. That might not matter for the question. At any rate, it seems legitimate for finite fields of order $p^e \gt p.$ | |
| Jan 2, 2018 at 20:15 | comment | added | Aaron Meyerowitz | Just to be specific, $1/(x(x+1))+1/((x+1)(x+2))+(p-2)/(x(x+2))=p/(x(x+2)).$ | |
| Jan 2, 2018 at 6:07 | comment | added | YCor | For $a_i\neq b_i$ These are $q(q-1)/2$ functions inside a subspace of dimension $q$... if $a,b,c$ are distinct, $1/((x-a)(x-b))$, $1/((x-b)(x-c))$, $1/((x-c)(x-a))$ are proportional to $1/(x-a)-1/(x-b)$, $1/(x-b)-1/(x-c)$, $1/(x-c)-1/(x-a)$ which have sum zero. So the answer is no as soon as $k,q\ge 3$. Did I miss something? | |
| Jan 2, 2018 at 5:58 | history | edited | user119164 | CC BY-SA 3.0 |
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| Jan 2, 2018 at 5:50 | history | edited | user119164 | CC BY-SA 3.0 |
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| Jan 2, 2018 at 5:49 | comment | added | abx | Not always: since $\dfrac{1}{(x+a)(x+a+1)} =\dfrac{1}{(x+a)}-\dfrac{1}{(x+a+1)}$, we have $\sum\limits_{a=0}^{p-1}\dfrac{1}{(x+a)(x+a+1)}=0\ $ over any field of characteristic $p$. | |
| Jan 2, 2018 at 5:45 | comment | added | user119164 | What if I assume I have far fewer than $p$ such functions? | |
| Jan 2, 2018 at 5:33 | review | First posts | |||
| Jan 2, 2018 at 7:11 | |||||
| Jan 2, 2018 at 5:29 | history | asked | user119164 | CC BY-SA 3.0 |