Timeline for Does sine interact equationally with addition alone?
Current License: CC BY-SA 4.0
28 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 15, 2021 at 21:54 | vote | accept | Noah Schweber | ||
| Jun 15, 2021 at 18:43 | comment | added | LSpice | @MattF.'s answer referenced above. | |
| Jun 15, 2021 at 17:12 | answer | added | user44143 | timeline score: 11 | |
| Jun 15, 2021 at 17:08 | comment | added | user44143 | @PaceNielsen, yes, I am writing that up now.... | |
| Jun 15, 2021 at 17:07 | comment | added | Pace Nielsen | @MattF. I wonder if that argument can't be slightly modified to work for sine. Since $\sin(z)$ is an analytic function, equality over $\mathbb{R}$ is equivalent to equality over $\mathbb{C}$. Now, rather than using "sufficiently large" one can use "on a circle, centered at the origin, in the complex plane, with sufficiently large radius", and compare the maximum of the absolute values of the two terms. | |
| Jun 15, 2021 at 15:30 | comment | added | user44143 | @PyRulez, no, subtraction is excluded. | |
| Jun 15, 2021 at 14:44 | comment | added | Christopher King | Is it implied that subtraction is also included? If so, you get that sin(x-x) = y-y. | |
| Jun 15, 2021 at 6:03 | comment | added | user44143 | @Gro-Tsen, I think this has a positive answer when $\sin$ is replaced by $\exp$. Suppose $t,u$ are terms in $L(+,\exp)$. Then we say $t<u$ iff $Qx_1 \ldots Qx_n t<u$ in the reals, where $Q$ means “for all sufficiently large”, and $x_1, \ldots x_n$ are the primitive variables in the language. The decision procedure for $t<u$ should be straightforward, and when neither $t<u$ nor $t>u$, it should be easy to prove $t=u$ from the rearrangements justified by $Th(+)$. | |
| Jun 15, 2021 at 1:37 | comment | added | Noah Schweber | @TimothyChow Ooh, that's a neat paper, thanks for pointing me towards it! (I also get the feeling that this should help give a positive answer to the question, and I also don't actually see how to do it.) | |
| Jun 14, 2021 at 23:09 | comment | added | Timothy Chow | It feels to me that the main result of Trigonometric diophantine equations by Conway and Jones should be relevant. I don't quite see how to connect all the dots, but it feels to me that we should be able to start with an alleged identity involving sin and chase through something like the Conway-Jones algorithm to arrive at an identity without sin. | |
| Jun 14, 2021 at 21:30 | comment | added | Noah Schweber | @Gro-Tsen I'm not sure how much I buy the connection with Schanuel. Do you see, for example, a way to resolve the exp-version of the question assuming Schanuel? I don't at a glance. | |
| Jun 14, 2021 at 19:29 | comment | added | Gro-Tsen | Do you know the answer to your question with “sin” replaced by “exp”? If not, it might be more natural to ask this (first). In any case, your question (at least with “sin” replaced by “exp”) seems strongly related to Schanuel's conjecture, which suggests it would be way out of reach of current techniques. | |
| Jun 14, 2021 at 18:42 | comment | added | Noah Schweber | @JoelDavidHamkins I mean, I had to google it :P. | |
| Jun 14, 2021 at 18:42 | comment | added | Joel David Hamkins | Oh dear, I don't know anything. | |
| Jun 14, 2021 at 18:40 | comment | added | Noah Schweber | @JoelDavidHamkins That never happens. | |
| Jun 14, 2021 at 18:39 | comment | added | Neil Strickland | @YCor Because $\sin(0)=0$ and $\sin(-x)=-\sin(x)$, it seems that the answer becomes negative as soon as we include zero or negation. | |
| Jun 14, 2021 at 18:39 | comment | added | Joel David Hamkins | Do we know any nontrivial rational values of $\sin$ at rational input values? That might seem likely to lead to a counterexample, right? Or perhaps this never happens? | |
| Jun 14, 2021 at 18:33 | comment | added | YCor | Out of curiosity, is there an obvious reason that allowing $-$ (minus sign) doesn't change the question? | |
| Jun 14, 2021 at 18:19 | comment | added | Joel David Hamkins | Ah, of course. But I do want $1$. | |
| Jun 14, 2021 at 18:17 | comment | added | Noah Schweber | @JoelDavidHamkins Re: your second question, no I don't (besides trivial observations such as "If we include $\pi$ as a constant we get the nontrivial equation $\sin(\pi)+\sin(\pi)=\sin(\pi)$"). | |
| Jun 14, 2021 at 18:16 | comment | added | Joel David Hamkins | Yes, that is what I had meant. So to refute the claim, it would suffice to exhibit a nontrivial identity satisfied by $\sin$ with $+$ in $\mathbb{R}$. | |
| Jun 14, 2021 at 18:16 | comment | added | Noah Schweber | @YCor I assume(d) Joel was folding in $+$ as you do. For full clarity: it's equivalent to asking whether every equation holding in $(\mathbb{R};+,\sin)$ also holds in $(\mathbb{R};+,f)$ for every unary function $f$. | |
| Jun 14, 2021 at 18:15 | comment | added | YCor | Isn't it rather that every identity follows from identities satisfied by $+$ in the reals? (would it be equivalent to: every identity $J(\sharp,u)$ such that $J(+,\sin)$ holds in the reals, is such that $J(+,f)$ holds for every self-map $f$ of the reals? | |
| Jun 14, 2021 at 18:15 | comment | added | Joel David Hamkins | Do you have counterexamples if we allow constants, such as $\pi$? Can we at least have $1$? | |
| Jun 14, 2021 at 18:05 | comment | added | Noah Schweber | @JoelDavidHamkins Yes, that's right. | |
| Jun 14, 2021 at 17:29 | comment | added | Joel David Hamkins | Is the question equivalent to asking whether $\sin $ only obeys trivial identities in the language with $+$, that is, only identities that every unary function would obey? | |
| Jun 14, 2021 at 17:08 | history | edited | LSpice | CC BY-SA 4.0 |
`\DeclareMathOperator`
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| Jun 14, 2021 at 16:50 | history | asked | Noah Schweber | CC BY-SA 4.0 |