Mathematics & Physics 2016, 9(3), 347–352 УДК 519.21 The Trigonometry of Harnack Curves Mikael Passare∗ Received 20.06.2016, received in revised form 20.06.2016, accepted 20.06.2016 Derive an explicit integral formula for the amoeba-to-coamoeba mapping in the case of polynomials that define Harnack curves. <...> As a consequence obtain an exact description of the coamoebas of such polynomials. <...> This formula can be viewed as a generalization of the familiar law of cosines that is used for solving triangles. <...> Keywords: Harnack curves, amoeba of polynomial, coamoeba of polynomial, Newton polygon, Ronkin function, law of cosines. <...> Introduction When studying Laurent expansions and Mellin transforms of multivariate rational functions, or equivalently Fourier series and Fourier transforms of rational functions in exponentials, one is naturally led to the concepts of amoebas and coamoebas. <...> The coamoeba A′f is defined similarly by means of the argument projection Arg(z) = (arg z1, . . . , arg zn). <...> Suppose we want to determine all Laurent series expansions (centered at the origin) of a rational function 1/f, where f is a polynomial. <...> It is then not hard to realize that there is one such expansion for each connected component of the amoeba complement Rn \ Af . <...> To each component of the amoeba complement there is associated an integer vector in the Newton polygon ∆f , called the order of the component, see [1]. <...> For ∗Mikael Passare (1959 – 2011) was a Professor in Mathematics at Stockholm University. <...> All rights reserved – 347 – Mikael Passare The Trigonometry of Harnack Curves each point (x, y) in the interior of the amoeba there are two points, conjugate to each other, on the complex line 1 − z − w = 0 that get mapped to (x, y) by the mapping Log. <...> The boundary points of Af correspond to real values of z and w = 1−z, and such points (z,w) are mapped by Arg to the vertices of the coamoeba A′f . <...> The composed mapping Arg ◦ Log−1 from the amoeba to the coamoeba, which is thus well defined up to sign, can easily be written down in this case. <...> Using the classical law of cosines we readily arrive at the formula Arg ◦ Log−1(x, y) = ± (arccos 1+e2x <...>