Timeline for Is there an analytic non-linear function that maps rational numbers to rational numbers and it maps irrational numbers to irrational numbers?
Current License: CC BY-SA 4.0
15 events
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Nov 16, 2022 at 0:44 | vote | accept | Francis Fan | ||
Nov 11, 2022 at 22:05 | comment | added | M.G. | @preferred_anon: it seems to me that your question boils down to the following: if we have some $P(x) \in \mathbb{Q}[x]$, say with only rational roots, can we perturb it by some $r \in \mathbb{Q}$ s.t. $Q(x) := P(x) - r$ has an irrational root? This is clearly so for quadratic $P$ ad hoc and is probably true for higher degree, too, but I don't have a good argument atm. | |
Nov 11, 2022 at 21:58 | comment | added | preferred_anon | Is it obvious that every polynomial of degree $>1$ with rational coefficients sends some irrational number to a rational number? A counterexample must have all rational roots, and I can't see how to get a rational out of an irrational, in that case. | |
Nov 11, 2022 at 18:37 | comment | added | M.G. | @AdamChalcraft: That's why I prefer to use "linear homogeneous" for emphasis, or homogeneous of degree 1. It's also more consistent from algebraic POV, but it's a mouthful. | |
Nov 11, 2022 at 17:39 | comment | added | Adam Chalcraft | @YCor It's annoying, but a degree-1 function is often called linear in the context of (constant, linear, quadratic, cubic, ...). Also in US schools when solving "simultaneous linear equations" using "linear algebra". The function $x\mapsto x+2$ is not linear but the polynomial $x+2$ is linear. Yuk. | |
Nov 11, 2022 at 14:36 | comment | added | M.G. | @Bumblebee: they have the same problem as a generic polynomial - can take irrational number to rational number, e.g. $(x^2+1)/(x^2-1)$ takes $\sqrt{2}$ to $3$. | |
Nov 11, 2022 at 4:17 | comment | added | Bumblebee | Don't rational functions answer your question? | |
Nov 10, 2022 at 13:05 | answer | added | Alexandre Eremenko | timeline score: 22 | |
Nov 10, 2022 at 10:54 | history | became hot network question | |||
Nov 10, 2022 at 10:01 | comment | added | Dave L Renfro | See the MSE question Functions that take rationals to rationals. | |
Nov 10, 2022 at 6:20 | comment | added | YCor | $x\mapsto kx+b$ is rather called an affine function than a linear function ($b=0$) | |
Nov 10, 2022 at 5:28 | history | edited | Daniele Tampieri | CC BY-SA 4.0 |
Math Jaxed+minor grammar improvements
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Nov 10, 2022 at 4:45 | answer | added | Saúl RM | timeline score: 16 | |
S Nov 10, 2022 at 2:50 | review | First questions | |||
Nov 10, 2022 at 3:49 | |||||
S Nov 10, 2022 at 2:50 | history | asked | Francis Fan | CC BY-SA 4.0 |