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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
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