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Hi,

In Remark 2 of "Zeros of Fekete Polynomials", (http://arxiv.org/PS_cache/math/pdf/9906/9906214v1.pdf), Conrey et. al. give $$\sup_{|z|=1}|f(z)| \ll p^{0.5} \log p.$$

But the Mahler measure of $f$, $M(f)$, is bounded from above by $\sup_{|z|=1}|f(z)|$. (I took this from (7.2) of "Experimental Number Theory" by F. R. Villegas.)

Since Mahler measure is multiplicative, then, letting $$ f = \prod_i f_i $$ where the product is over all irreducible factors, one has $$ M(f) = \prod_i M(f_i). $$

Using lower bounds for Mahler measures found in "The Mahler measure of Algebraic Numbers: A Survey" by C. Smyth,(http://www.maths.ed.ac.uk/~chris/Smyth240707.pdf), perhaps one can, comparing upper and lower bounds for Mahler measures, get a nontrivial upper bound on the number of irreducible factors of the Fekete polynomial? For example, (11) of the paper by Smyth gives, for the algebraic number $\alpha$ with minimal polynomial of degree $d \geq 2$, $$ M(\alpha) > 1 + \frac{1}{1200}\left(\frac{\log \log d}{\log d}\right)^3. $$

Hi,

In Remark 2 of "Zeros of Fekete Polynomials", (http://arxiv.org/PS_cache/math/pdf/9906/9906214v1.pdf), Conrey et. al. give $$\sup_{|z|=1}|f(z)| \ll p^{0.5} \log p.$$

But the Mahler measure of $f$, $M(f)$, is bounded from above by $\sup_{|z|=1}|f(z)|$. (I took this from (7.4) of "Experimental Number Theory" by F. R. Villegas.)

Since Mahler measure is multiplicative, then, letting $$ f = \prod_i f_i $$ where the product is over all irreducible factors, one has $$ M(f) = \prod_i M(f_i). $$

Using lower bounds for Mahler measures found in "The Mahler measure of Algebraic Numbers: A Survey" by C. Smyth,(http://www.maths.ed.ac.uk/~chris/Smyth240707.pdf), perhaps one can, comparing upper and lower bounds for Mahler measures, get a nontrivial upper bound on the number of irreducible factors of the Fekete polynomial? For example, (11) of the paper by Smyth gives, for the algebraic number $\alpha$ with minimal polynomial of degree $d \geq 2$, $$ M(\alpha) > 1 + \frac{1}{1200}\left(\frac{\log \log d}{\log d}\right)^3. $$

Hi,

In Remark 2 of "Zeros of Fekete Polynomials", (http://arxiv.org/PS_cache/math/pdf/9906/9906214v1.pdf), Conrey et. al. give $$\sup_{|z|=1}|f(z)| \ll p^{0.5} \log p$$.

But the Mahler measure of $f$, $M(f)$, is bounded from above by $\sup_{|z|=1}|f(z)|$. (I took this from (7.2) of "Experimental Number Theory by F. R. Villegas.)

Since Mahler measure is multiplicative, then, letting $$ f = \prod_i f_i $$ where the product is over all irreducible factors, one has $$ M(f) = \prod_i M(f_i). $$

Using lower bounds for Mahler measures found in "The Mahler measure of Algebraic Numbers: A Survey" by C. Smyth,(http://www.maths.ed.ac.uk/~chris/Smyth240707.pdf), perhaps one can, comparing upper and lower bounds for Mahler measures, get a nontrivial upper bound on the number of irreducible factors of the Fekete polynomial? For example, (11) of the paper by Smyth gives, for the algebraic number $\alpha$ with minimal polynomial of degree $d \geq 2$, $$ M(\alpha) > 1 + \frac{1}{1200}\left(\frac{\log \log d}{\log d}\right)^3. $$

Hi,

In Remark 2 of "Zeros of Fekete Polynomials", (http://arxiv.org/PS_cache/math/pdf/9906/9906214v1.pdf), Conrey et. al. give $$\sup_{|z|=1}|f(z)| \ll p^{0.5} \log p.$$

But the Mahler measure of $f$, $M(f)$, is bounded from above by $\sup_{|z|=1}|f(z)|$. (I took this from (7.2) of "Experimental Number Theory" by F. R. Villegas.)

Since Mahler measure is multiplicative, then, letting $$ f = \prod_i f_i $$ where the product is over all irreducible factors, one has $$ M(f) = \prod_i M(f_i). $$

Using lower bounds for Mahler measures found in "The Mahler measure of Algebraic Numbers: A Survey" by C. Smyth,(http://www.maths.ed.ac.uk/~chris/Smyth240707.pdf), perhaps one can, comparing upper and lower bounds for Mahler measures, get a nontrivial upper bound on the number of irreducible factors of the Fekete polynomial? For example, (11) of the paper by Smyth gives, for the algebraic number $\alpha$ with minimal polynomial of degree $d \geq 2$, $$ M(\alpha) > 1 + \frac{1}{1200}\left(\frac{\log \log d}{\log d}\right)^3. $$

Hi,

In Remark 2 of "Zeros of Fekete Polynomials", (http://arxiv.org/PS_cache/math/pdf/9906/9906214v1.pdf), Conrey et. al. give $$\sup_{|z|=1}|f(z)| \ll p^{0.5} \log p$$.

But the Mahler measure of $f$, $M(f)$, is bounded from above by $\sup_{|z|=1}|f(z)|$. (I took this from (7.2) of "Experimental Number Theory by F. R. Villegas.)

Since Mahler measure is multiplicative, then, letting $$ f = \prod_i f_i $$ where the product is over all irreducible factors, one has $$ M(f) = \prod_i M(f_i). $$

Using lower bounds for Mahler measures found in "The Mahler measure of Algebraic Numbers: A Survey" by C. Smyth,(http://www.maths.ed.ac.uk/~chris/Smyth240707.pdf), perhaps one can, comparing upper and lower bounds for Mahler measures, get a nontrivial upper bound on the number of irreducible factors of the Fekete polynomial? For example, (11) of the paper by Smyth gives, for the algebraic number $\alpha$ with minimal polynomial of degree $d$, $$ M(\alpha) > 1 + \frac{1}{1200}\left(\frac{\log \log d}{\log d}\right)^3. $$

Hi,

In Remark 2 of "Zeros of Fekete Polynomials", (http://arxiv.org/PS_cache/math/pdf/9906/9906214v1.pdf), Conrey et. al. give $$\sup_{|z|=1}|f(z)| \ll p^{0.5} \log p$$.

But the Mahler measure of $f$, $M(f)$, is bounded from above by $\sup_{|z|=1}|f(z)|$. (I took this from (7.2) of "Experimental Number Theory by F. R. Villegas.)

Since Mahler measure is multiplicative, then, letting $$ f = \prod_i f_i $$ where the product is over all irreducible factors, one has $$ M(f) = \prod_i M(f_i). $$

Using lower bounds for Mahler measures found in "The Mahler measure of Algebraic Numbers: A Survey" by C. Smyth,(http://www.maths.ed.ac.uk/~chris/Smyth240707.pdf), perhaps one can, comparing upper and lower bounds for Mahler measures, get a nontrivial upper bound on the number of irreducible factors of the Fekete polynomial? For example, (11) of the paper by Smyth gives, for the algebraic number $\alpha$ with minimal polynomial of degree $d \geq 2$, $$ M(\alpha) > 1 + \frac{1}{1200}\left(\frac{\log \log d}{\log d}\right)^3. $$