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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There are many articles in the literature that give effective forms of the Nullstellensatz, including explicit bounds for the degrees and for the heights of the polynomials used to form the various linear combination(s). These results have been extensively used in transcendence theory. The following paper gives strong bounds on the degrees: Kollár, J.: Sharp effective Nullstellensatz. J. Am. Math. Soc. 1, 963–975 (1988) MR0944576 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 |
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