## Are there known quantitative descriptions of the fact that the common zero set of some polynomials is empty besides Nullstellensatz?

Working in a polynomial ring, if some polynomials has no common zeros, Nullstellensatz tells us a qualitative description of the propertity of these polynomials, are there known quantitative descriptions? For example, these descriptions may tell us information on the degrees or something else of the polynomials.

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For a bound on the coefficients of the polynomials (at least over $\mathbb{Q}$), see for example Chapter 4 of: Masser, D. W.; Wüstholz, G.: Fields of large transcendence degree generated by values of elliptic functions. Invent. Math. 72 (1983), no. 3, 407–464. MR0704399