Timeline for answer to Positive roots of real exponents function by Iosif Pinelis
Current License: CC BY-SA 4.0
Post Revisions
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 8, 2025 at 15:52 | vote | accept | VSP | ||
| Oct 26, 2025 at 2:52 | comment | added | Iosif Pinelis | @VSP : Your post did not say anything about allowed methods. It also did not mention more general settings. If this method does not work in a more general case, unstated in your post, it is all the more of a reason to post the more general questions separately. That would be better in a number of respects. One of them would be giving better exposure to what is really of interest to you. | |
| Oct 26, 2025 at 2:40 | history | edited | Iosif Pinelis | CC BY-SA 4.0 |
added 187 characters in body
|
| Oct 26, 2025 at 2:34 | comment | added | VSP | To use this proof we must know $h_{12}(x)$, $h_{13}(x)$ and so on. For more terms case, its difficult to find these functions. That’s why this proof is not that useful. | |
| Oct 26, 2025 at 2:25 | comment | added | Iosif Pinelis | @VSP : The same idea should work, in principle, for any number of terms. Taking $f'$, we have $f'(x)=bg_{11}(x)+cg_{12}(x)+dg_{13}(x)$ for certain function $g_{11},g_{12},g_{13}$. Dividing both sides of this equation by $g_{11}(x)$, we get $f_1(x)=b+ch_{12}(x)+dh_{13}(x)$, thus killing $a$ and getting three remaining coefficients $b,c,d$ instead of the initial four coefficients $a,b,c,d$. Continuing thus, we kill all the coefficients $a,b,c,d$ and get $f_3$. Anyhow, your question has been fully answered. If you have additional questions, consider posting them separately. | |
| Oct 26, 2025 at 2:06 | comment | added | VSP | @losif, how are you constructing $f_2(x)$ and $f_3(x)$? I want to prove it in general. This idea won't work for the case of five term and six term cases. I want to prove it by induction (by approach 3). We have $f(x) = a+(x+3)^rg(x)$. Can we show the number of positive zeros of $f(x)$ is not more than that of $g(x)$? | |
| Oct 26, 2025 at 1:55 | history | edited | Iosif Pinelis | CC BY-SA 4.0 |
added 194 characters in body
|
| Oct 26, 2025 at 1:46 | history | edited | Iosif Pinelis | CC BY-SA 4.0 |
added 407 characters in body
|
| Oct 26, 2025 at 1:27 | history | edited | Iosif Pinelis | CC BY-SA 4.0 |
added 617 characters in body
|
| Oct 25, 2025 at 23:29 | history | answered | Iosif Pinelis | CC BY-SA 4.0 |