Geodesic
Multiplicity estimates for theta constants
Fundamentalʹnaâ i prikladnaâ matematika, Tome 5 (1999) no. 2, pp. 557-562
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An upper bound for multiplicity of zero at the point $q=0$ for polynomials in logarithmic derivatives of theta constants is proved. The upper bound depends on degrees of the polynomials. The proof is based on a description in terms of theta constants of the general solution of a system of algebraic differential equations. 
@article{FPM_1999_5_2_a11,
     author = {Yu. V. Nesterenko},
     title = {Multiplicity estimates for theta constants},
     journal = {Fundamentalʹna\^a i prikladna\^a matematika},
     pages = {557--562},
     year = {1999},
     volume = {5},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FPM_1999_5_2_a11/}
}
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Yu. V. Nesterenko. Multiplicity estimates for theta constants. Fundamentalʹnaâ i prikladnaâ matematika, Tome 5 (1999) no. 2, pp. 557-562. http://geodesic.mathdoc.fr/item/FPM_1999_5_2_a11/