Mixing
Existence of a monic polynomial with integer coefficients and a given set of root
2
$begingroup$
Let $r > 1$, $epsilon > 0$, $eta > 0$ does there always exist a monic polynomial with integer coefficients $P$ such that
$P$ has a unique real root $r_0$, s.t $|r_0 - r| < epsilon$
- For all other roots of $P$, $r_i$, $|r_i| < eta$
(I'm expecting this to be false)
polynomials
asked Jan 23 at 17:21
Arthur B.Arthur B.
442212
$endgroup$
add a comment |
2
$begingroup$
Let $r > 1$, $epsilon > 0$, $eta > 0$ does there always exist a monic polynomial with integer coefficients $P$ such that
$P$ has a unique real root $r_0$, s.t $|r_0 - r| < epsilon$
- For all other roots of $P$, $r_i$, $|r_i| < eta$
(I'm expecting this to be false)
polynomials
asked Jan 23 at 17:21
Arthur B.Arthur B.
442212
$endgroup$
$begingroup$
Once the degree $n$ is fixed, those bounds on the roots give some bounds on the coefficients of the polynomial, so there are only finitely many integer polynomials satisfying those. Thus for $r$ non-algebraic when $epsilon to 0$ then necessarily $n to infty$. Also (with the same argument) for $eta$ small enough there is no monic integer polynomial whose roots are all $eta$ close to $0$, so wlog. your polynomial must be irreducible
$endgroup$
– reuns
Jan 25 at 3:22
add a comment |
2
2
2
1
$begingroup$
Let $r > 1$, $epsilon > 0$, $eta > 0$ does there always exist a monic polynomial with integer coefficients $P$ such that
$P$ has a unique real root $r_0$, s.t $|r_0 - r| < epsilon$
- For all other roots of $P$, $r_i$, $|r_i| < eta$
(I'm expecting this to be false)
polynomials
asked Jan 23 at 17:21
Arthur B.Arthur B.
442212
$endgroup$
Let $r > 1$, $epsilon > 0$, $eta > 0$ does there always exist a monic polynomial with integer coefficients $P$ such that
$P$ has a unique real root $r_0$, s.t $|r_0 - r| < epsilon$
- For all other roots of $P$, $r_i$, $|r_i| < eta$
(I'm expecting this to be false)
polynomials
polynomials
asked Jan 23 at 17:21
Arthur B.Arthur B.
442212
asked Jan 23 at 17:21
Arthur B.Arthur B.
442212
edited Jan 24 at 5:08
Arthur B.
asked Jan 23 at 17:21
Arthur B.Arthur B.
442212
asked Jan 23 at 17:21
Arthur B.Arthur B.
442212
asked Jan 23 at 17:21
Arthur B.Arthur B.
442212
442212
$begingroup$
Once the degree $n$ is fixed, those bounds on the roots give some bounds on the coefficients of the polynomial, so there are only finitely many integer polynomials satisfying those. Thus for $r$ non-algebraic when $epsilon to 0$ then necessarily $n to infty$. Also (with the same argument) for $eta$ small enough there is no monic integer polynomial whose roots are all $eta$ close to $0$, so wlog. your polynomial must be irreducible
$endgroup$
– reuns
Jan 25 at 3:22
add a comment |
$begingroup$
Once the degree $n$ is fixed, those bounds on the roots give some bounds on the coefficients of the polynomial, so there are only finitely many integer polynomials satisfying those. Thus for $r$ non-algebraic when $epsilon to 0$ then necessarily $n to infty$. Also (with the same argument) for $eta$ small enough there is no monic integer polynomial whose roots are all $eta$ close to $0$, so wlog. your polynomial must be irreducible
$endgroup$
– reuns
Jan 25 at 3:22
$begingroup$
Once the degree $n$ is fixed, those bounds on the roots give some bounds on the coefficients of the polynomial, so there are only finitely many integer polynomials satisfying those. Thus for $r$ non-algebraic when $epsilon to 0$ then necessarily $n to infty$. Also (with the same argument) for $eta$ small enough there is no monic integer polynomial whose roots are all $eta$ close to $0$, so wlog. your polynomial must be irreducible
$endgroup$
– reuns
Jan 25 at 3:22
$begingroup$
Once the degree $n$ is fixed, those bounds on the roots give some bounds on the coefficients of the polynomial, so there are only finitely many integer polynomials satisfying those. Thus for $r$ non-algebraic when $epsilon to 0$ then necessarily $n to infty$. Also (with the same argument) for $eta$ small enough there is no monic integer polynomial whose roots are all $eta$ close to $0$, so wlog. your polynomial must be irreducible
$endgroup$
– reuns
Jan 25 at 3:22
add a comment |
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$begingroup$
Once the degree $n$ is fixed, those bounds on the roots give some bounds on the coefficients of the polynomial, so there are only finitely many integer polynomials satisfying those. Thus for $r$ non-algebraic when $epsilon to 0$ then necessarily $n to infty$. Also (with the same argument) for $eta$ small enough there is no monic integer polynomial whose roots are all $eta$ close to $0$, so wlog. your polynomial must be irreducible
$endgroup$
– reuns
Jan 25 at 3:22