Timeline for Testing whether $e^x+ax^2+bx+c$ has a zero
Current License: CC BY-SA 4.0
16 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 1, 2021 at 1:30 | vote | accept | CommunityBot | moved from User.Id=44143 by developer User.Id=481663 | |
| Mar 23, 2021 at 22:19 | comment | added | user44143 | This looks good to me. I will wait for other answers until the bounty expires on the question linked at the beginning, and assuming I don't get any, I will accept this. In any case, if you like this sort of analysis, a decision procedure for the existential theory of the reals under exponentiation, and one whose correctness does not depend on Schanuel's conjecture, would be very nice. | |
| Mar 23, 2021 at 15:10 | history | edited | Max Alekseyev | CC BY-SA 4.0 |
cosmetic changes
|
| Mar 23, 2021 at 12:24 | history | edited | Max Alekseyev | CC BY-SA 4.0 |
deleted 28 characters in body
|
| Mar 23, 2021 at 11:38 | comment | added | user44143 | Curiously, this leads to explicit expressions which are not exponential polynomials, but rather things like $s(\ln(-s)-1)+t\le 0$, which are expressions in $a,b,c$ with lots of divisions and square roots. We can convert the inequality with one $\ln$ into an inequality with one $\exp$, but there will still be square roots inside the $\exp$ that I don’t see how to eliminate. So if the best we can do is to formulate criteria as a mix of exponential and algebraic functions, that is a significant difference from the quantifier elimination for the reals without exponentiation. | |
| Mar 23, 2021 at 11:36 | history | edited | Max Alekseyev | CC BY-SA 4.0 |
added 60 characters in body
|
| Mar 23, 2021 at 11:25 | history | edited | user44143 | CC BY-SA 4.0 |
added 178 characters in body
|
| Mar 23, 2021 at 3:59 | history | edited | Max Alekseyev | CC BY-SA 4.0 |
added 11 characters in body
|
| Mar 23, 2021 at 3:50 | history | edited | Max Alekseyev | CC BY-SA 4.0 |
remark
|
| Mar 23, 2021 at 3:42 | comment | added | Max Alekseyev | @MattF.: Thanks for streamlining my answer. As for the last two subcases, I'm not sure what exactly is unclear. Just as an example, if in last case $\text{TEST}(2a,b,s_0,t)$ is True, $f'(x)$ has a zero $x=z$ in the interval $(s_0,t]\subset (0,t]\cap (s,t]$. Then $z>0$ and $f(z)\leq 0$, implying the Yes answer. | |
| Mar 23, 2021 at 2:27 | history | edited | user44143 | CC BY-SA 4.0 |
gave names to roots of quadratic
|
| Mar 23, 2021 at 1:55 | history | edited | user44143 | CC BY-SA 4.0 |
made assumptions easier to follow
|
| Mar 22, 2021 at 22:45 | history | edited | Max Alekseyev | CC BY-SA 4.0 |
added 2 characters in body
|
| Mar 22, 2021 at 22:26 | history | edited | Max Alekseyev | CC BY-SA 4.0 |
added 254 characters in body; deleted 10 characters in body
|
| Mar 22, 2021 at 22:07 | history | edited | Max Alekseyev | CC BY-SA 4.0 |
added 42 characters in body
|
| Mar 22, 2021 at 21:57 | history | answered | Max Alekseyev | CC BY-SA 4.0 |