Abstract
The first section of this chapter gives general constructive versions of the Nullstellensatz for a polynomial system over a discrete field (we will be able to compare Theorems 1.5 (p. 384), 1.8 (p. 386) and 1.9 (p. 386), to Theorems III-9.5 (p. 133) and III-9.7 (p. 134)). We also give a simultaneous Noether positioning theorem (Theorem 1.7).
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- 1.
On this second point, our assertion is less clear. If we return to the example of the decomposition of a polynomial into prime factors, it is impossible to achieve the result algorithmically over certain fields.
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It would be more natural to say: fundamental system of orthogonal elements.
- 3.
For a commutative ring \(\mathbf {k}\), \(\mathop {\mathsf {Zar}}\nolimits \mathbf {k}\) is the set of radicals of finitely generated ideals of \(\mathbf {k}\) (Sect. XI-4). It is a distributive lattice. In classical mathematics, \(\mathop {\mathsf {Zar}}\nolimits \mathbf {k}\) is identified with the lattice of quasi-compact open sets of the spectral space \(\mathop {\mathsf {Spec}}\nolimits \mathbf {k}\) (Sect. XIII-1).
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It follows that if \(\mathbf {K}\) is a general field (see Sect. IX-1), the questions of computability are actually discussed entirely in \(\mathop {\mathrm {Frac}}\nolimits (\mathbf {Z})=\mathop {\mathrm {Frac}}\nolimits (\mathbf {Z}_0)\otimes _{\mathbf {Z}_0}\mathbf {Z}={(\mathbf {Z}_0^\star )^{-1}}\mathbf {Z}\), and \(\mathop {\mathrm {Frac}}\nolimits (\mathbf {Z})\) is discrete if \(\mathbf {Z}_0\) is itself a discrete ring. As \(\mathbf {Z}_0\) is a finitely generated ring, it certainly is, in classical mathematics, an effective (also called computable) ring with an explicit equality test, in the sense of recursion theory via Turing machines.
But this last result is not a truly satisfying approach to the reality of the computation. It is indeed akin to results in classical mathematics of the form “every recursive real number admits a recursive development into a continued fraction”, a theorem that is evidently false from a practical point of view, since to implement it, one must first know whether the number is rational or not.
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This quote translates as: “Given an algebraic equation, that we consider as replaced by the system \((S)\) of relations between the roots \(x_1,\ldots ,x_n\) and the coefficients, we first study the following fundamental problem: What subset can we extract from the knowledge of certain relations \((A)\) between \(x_1,\ldots ,x_n\), by only employing rational operations? We show that we can deduce from the system \((S,A)\) an analogous system, for which the system \((S,A)\) admits all the solutions, and which is, as we say, automorphic: which means that its diverse solutions are deduced from any one of them by the substitutions of a group \(G\), which is said to be the group associated with the system, or simply the group of the system. We will notice that \(S\) is already an automorphic system, with the general group being its associated group. From then on, if we take Galois’ point of view, ... we see that we can limit ourselves to only considering rational and automorphic systems \((S,A)\).”
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© 2015 Springer Science+Business Media Dordrecht
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Lombardi, H., Quitté, C. (2015). The Dynamic Method. In: Commutative Algebra: Constructive Methods. Algebra and Applications, vol 20. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-9944-7_7
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DOI: https://doi.org/10.1007/978-94-017-9944-7_7
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