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user_lambda
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Consider a polynomial $\sum\limits_{k=0}^n a_kx^k$ with $a_k\geq 0$ and $x\geq 0$. In this comment, Richard Stanley mentions that polynomials with only real roots are log concave functions. Can somebody provide a reference for this result? I can't find it anywhere. I am in particular interested to know if a similar result holds for multivariate polynomials.

Also, does anybody know other results about the log-concavity of polynomials as functions?

(To be clear I'm not talking about log-concave polynomials in the sense that their coefficients form a log-concave sequence.)

Edit: I've added restrictions such that the polynomial is never negative.

Consider a polynomial $\sum\limits_{k=0}^n a_kx^k$ with $a_k\geq 0$ and $x\geq 0$. In this comment, Richard Stanley mentions that polynomials with only real roots are log concave functions. Can somebody provide a reference for this result? I can't find it anywhere. I am in particular interested to know if a similar result holds for multivariate polynomials.

Also, does anybody know other results about the log-concavity of polynomials as functions?

(To be clear I'm not talking about log-concave polynomials in the sense that their coefficients form a log-concave sequence.)

Consider a polynomial $\sum\limits_{k=0}^n a_kx^k$ with $a_k\geq 0$ and $x\geq 0$. In this comment, Richard Stanley mentions that polynomials with only real roots are log concave functions. Can somebody provide a reference for this result? I can't find it anywhere. I am in particular interested to know if a similar result holds for multivariate polynomials.

Also, does anybody know other results about the log-concavity of polynomials as functions?

(To be clear I'm not talking about log-concave polynomials in the sense that their coefficients form a log-concave sequence.)

Edit: I've added restrictions such that the polynomial is never negative.

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user_lambda
user_lambda
  • 389
  • 1
  • 6

InConsider a polynomial $\sum\limits_{k=0}^n a_kx^k$ with $a_k\geq 0$ and $x\geq 0$. In this comment, Richard Stanley mentions that polynomials with only real roots are log concave functions. Can somebody provide a reference for this result? I can't find it anywhere. I am in particular interested to know if a similar result holds for multivariate polynomials.

Also, does anybody know other results about the log-concavity of polynomials as functions?

(To be clear I'm not talking about log-concave polynomials in the sense that their coefficients form a log-concave sequence.)

In this comment, Richard Stanley mentions that polynomials with only real roots are log concave functions. Can somebody provide a reference for this result? I can't find it anywhere. I am in particular interested to know if a similar result holds for multivariate polynomials.

Also, does anybody know other results about the log-concavity of polynomials as functions?

(To be clear I'm not talking about log-concave polynomials in the sense that their coefficients form a log-concave sequence.)

Consider a polynomial $\sum\limits_{k=0}^n a_kx^k$ with $a_k\geq 0$ and $x\geq 0$. In this comment, Richard Stanley mentions that polynomials with only real roots are log concave functions. Can somebody provide a reference for this result? I can't find it anywhere. I am in particular interested to know if a similar result holds for multivariate polynomials.

Also, does anybody know other results about the log-concavity of polynomials as functions?

(To be clear I'm not talking about log-concave polynomials in the sense that their coefficients form a log-concave sequence.)

Source Link
user_lambda
user_lambda
  • 389
  • 1
  • 6

Are polynomials with only real zeros log concave functions?

In this comment, Richard Stanley mentions that polynomials with only real roots are log concave functions. Can somebody provide a reference for this result? I can't find it anywhere. I am in particular interested to know if a similar result holds for multivariate polynomials.

Also, does anybody know other results about the log-concavity of polynomials as functions?

(To be clear I'm not talking about log-concave polynomials in the sense that their coefficients form a log-concave sequence.)