Skip to main content
MathOverflow

Return to Question

added 4 characters in body
Source Link
Dr J
Dr J
  • 145
  • 5

Consider a large, fixed $M>2$. For each $n$, let $\zeta_n$$\alpha_n$ denote the smallest algebraic integer of degree at most $n$, all of whose Galois conjugates lie in the real interval $(0,M)$.

Is there anything known on the rate at which $\zeta_n$$\alpha_n$ decays to $0$ as $n\to\infty$ ?

The exponential lower-bound $\zeta_n>\frac1 {M^{n}}$$\alpha_n>\frac1 {M^{n}}$ is not hard to prove, but is it reasonable to expect that $\zeta_n$$\alpha_n$ will actually decay much more slowly, i.e. like an inverse power of $n$ ?

Consider a large, fixed $M>2$. For each $n$, let $\zeta_n$ denote the smallest algebraic integer of degree at most $n$, all of whose Galois conjugates lie in the real interval $(0,M)$.

Is there anything known on the rate at which $\zeta_n$ decays to $0$ as $n\to\infty$ ?

The exponential lower-bound $\zeta_n>\frac1 {M^{n}}$ is not hard to prove, but is it reasonable to expect that $\zeta_n$ will actually decay much more slowly, i.e. like an inverse power of $n$ ?

Consider a large, fixed $M>2$. For each $n$, let $\alpha_n$ denote the smallest algebraic integer of degree at most $n$, all of whose Galois conjugates lie in the real interval $(0,M)$.

Is there anything known on the rate at which $\alpha_n$ decays to $0$ as $n\to\infty$ ?

The exponential lower-bound $\alpha_n>\frac1 {M^{n}}$ is not hard to prove, but is it reasonable to expect that $\alpha_n$ will actually decay much more slowly, i.e. like an inverse power of $n$ ?

Source Link
Dr J
Dr J
  • 145
  • 5

How small can a totally positive integer be?

Consider a large, fixed $M>2$. For each $n$, let $\zeta_n$ denote the smallest algebraic integer of degree at most $n$, all of whose Galois conjugates lie in the real interval $(0,M)$.

Is there anything known on the rate at which $\zeta_n$ decays to $0$ as $n\to\infty$ ?

The exponential lower-bound $\zeta_n>\frac1 {M^{n}}$ is not hard to prove, but is it reasonable to expect that $\zeta_n$ will actually decay much more slowly, i.e. like an inverse power of $n$ ?