## Identities of Mahler measures of certain polynomials defining curves of arbitrary genus.(English)Zbl 1467.11095

In this paper, the authors give some identities for Mahler measures of polynomials in two variables. For example, in Theorem 2, defining $$P_k(x,y)=y^2+(kx+1)y+x^3$$, they show that $2m(P_k(x-1/k,y)) = 3m(P_k(x,y))-\log |k|$ for $$|k| \geq k_0=3.2407\dots$$, where $$k_0$$ is the root of some quartic polynomial. Similar identity $m(Q_k(x-1/k,y)) = 2m(Q_k(x,y))-\log |k|$ is obtained for the shift of the polynomial $$Q_k(x,y)=y^2+(2x^2+kx+1)y+x^4$$ for $$|k| \geq k_1=5.9794\dots$$, where $$k_1$$ is the root of some quintic polynomial.

### MSC:

 11R06 PV-numbers and generalizations; other special algebraic numbers; Mahler measure 11R09 Polynomials (irreducibility, etc.) 19F27 Étale cohomology, higher regulators, zeta and $$L$$-functions ($$K$$-theoretic aspects)
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### References:

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