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Sharp estimates for the arithmetic Nullstellensatz Item Preview. EMBED for wordpress. Want more? Advanced embedding details, examples, and help! We also present degree and height estimates of intrinsic type, which depend mainly on the degree and the height of the input polynomial system. As an application, we derive an effective arithmetic Nullstellensatz for sparse polynomial systems.
The proof of these results relies heavily on the notion of local height of an affine variety defined over a number field. We introduce this notion and study its basic properties.
MathOverflow is a question and answer site for professional mathematicians. It only takes a minute to sign up. I would like to know the general form of this result, and see some reference where I can read about it and some applications perhaps. All help is appreciated.
Sign up to join this community. The best answers are voted up and rise to the top. Home Questions Tags Users Unanswered. What is the arithmetic Nullstellensatz? Ask Question.
Asked 6 years, 1 month ago. Active 4 years, 4 months ago. Viewed 1k times. Jose Brox 2, 2 2 gold badges 32 32 silver badges 45 45 bronze badges.
Polynomial ring Polynomial ring 3 3 bronze badges. Rhetorical question, yes, but can your proposed counterexample be fixed? Active Oldest Votes. Dietrich Burde Dietrich Burde I am somehow curious whether there is some big picture result that generalizes the result from Gowers book.
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Sharp estimates for the arithmetic Nullstellensatz
It only takes a minute to sign up. Background: I have just started learning basic algebraic geometry. My solution to a simple problem involves an application of the Nullstellensatz and I want to know whether this is overkill or perhaps plainly wrong. This is just what I need. This is one of the maps to do so, the inverse map is still needed. This isn't particularly difficult. This would exhibit an isomorphism. Yes, one can avoid the use of Nullstellensatz. Then we have two obviuous facts:.
Verification of statement 1. Verification of statement 2. Sign up to join this community. The best answers are voted up and rise to the top.
Home Questions Tags Users Unanswered. Basic application of the Nullstellensatz Ask Question. Asked 5 years, 4 months ago. Active 5 years, 4 months ago. Viewed times. Is my argument wrong, or is there a much easier way to answer this question?Hilbert's Nullstellensatz German for "theorem of zeros," or more literally, "zero-locus-theorem"—see Satz is a theorem that establishes a fundamental relationship between geometry and algebra.
This relationship is the basis of algebraic geometrya branch of mathematics. It relates algebraic sets to ideals in polynomial rings over algebraically closed fields. This relationship was discovered by David Hilbert who proved the Nullstellensatz and several other important related theorems named after him like Hilbert's basis theorem.
This is the reason for the name of the theorem, which can be proved easily from the 'weak' form using the Rabinowitsch trick. In this way, we obtain an order-reversing bijective correspondence between the algebraic sets in K n and the radical ideals of K [ X 1…X n ].Commutative algebra 31 (Nullstellensatz)
More generally. There are many known proofs of the theorem. One proof uses Zariski's lemmawhich asserts that, if a field is finitely generated as an associative algebra over a field kthen it is a finite field extension of k that is, it is also finitely generated as a vector space.
Here is a sketch of this proof. The Nullstellensatz will also follow trivially once one systematically developed the theory of a Jacobson ringa ring in which a radical ideal is an intersection of maximal ideals. In all of its variants, Hilbert's Nullstellensatz asserts that some polynomial g belongs or not to an ideal generated, say, by f 1This means the existence or the non-existence of polynomials g 1The usual proofs of the Nullstellensatz are not constructive, non-effective, in the sense that they do not give any way to compute the g i.
It is thus a rather natural question to ask if there is an effective way to compute the g i and the exponent r in the strong form or to prove that they do not exist.
To solve this problem, it suffices to provide an upper bound on the total degree of the g i : such a bound reduces the problem to a finite system of linear equations that may be solved by usual linear algebra techniques. Any such upper bound is called an effective Nullstellensatz.
A related problem is the ideal membership problemwhich consists in testing if a polynomial belongs to an ideal. For this problem also, a solution is provided by an upper bound on the degree of the g i. A general solution of the ideal membership problem provides an effective Nullstellensatz, at least for the weak form. InGrete Hermann gave an upper bound for ideal membership problem that is doubly exponential in the number of variables.
In Mayr and Meyer gave an example where the g i have a degree that is at least double exponential, showing that every general upper bound for the ideal membership problem is doubly exponential in the number of variables.An effective Nullstellensatz gives information on some aspect of the complexity of such a representation.
Although classical work of G. In the various models of computation, how many steps might be involved in finding such a representation?
In these models, as in the above classical case for heights, the complexity bounds turn out to be better than that obtained by simply applying linear algebra and using the degree bounds. Many interesting aspects, such as sparsity of the coefficients, arise. See [a9] for an introduction to a sizable and growing literature. Various effective generalizations of Hilbert's Nullstellensatz exist, cf.
Log in. Namespaces Page Discussion. Views View View source History. Jump to: navigationsearch. Encyclopedia of Mathematics. This article was adapted from an original article by W. See original article.
Categories : TeX semi-auto TeX partially done. This page was last edited on 1 Julyat Blum, F. Cuker, M. Schub, S. Smale, "Complexity and real computation"Springer Berenstein, A.
Yger, "Residue calculus and effective Nullstellensatz" Amer. Caniglia, A.However, what makes this factors difficult to measure in search engines is the lack of data supporting it. User engagement is too indirect an influence on search rankings.
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