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Hilbert's basis theorem

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In mathematics, specifically commutative algebra, Hilbert's basis theorem says that a polynomial ring over a Noetherian ring is Noetherian.

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Statement

If is a ring, let denote the ring of polynomials in the indeterminate over . Hilbert proved that if is "not too large", in the sense that if is Noetherian, the same must be true for . Formally,

Hilbert's Basis Theorem. If is a Noetherian ring, then is a Noetherian ring.

Corollary. If is a Noetherian ring, then is a Noetherian ring.

This can be translated into algebraic geometry as follows: every algebraic set over a field can be described as the set of common roots of finitely many polynomial equations. Hilbert proved the theorem (for the special case of polynomial rings over a field) in the course of his proof of finite generation of rings of invariants.[1]

Hilbert produced an innovative proof by contradiction using mathematical induction; his method does not give an algorithm to produce the finitely many basis polynomials for a given ideal: it only shows that they must exist. One can determine basis polynomials using the method of Gröbner bases.

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Ring (mathematics)

Ring (mathematics)

In mathematics, rings are algebraic structures that generalize fields: multiplication need not be commutative and multiplicative inverses need not exist. In other words, a ring is a set equipped with two binary operations satisfying properties analogous to those of addition and multiplication of integers. Ring elements may be numbers such as integers or complex numbers, but they may also be non-numerical objects such as polynomials, square matrices, functions, and power series.

Polynomial

Polynomial

In mathematics, a polynomial is an expression consisting of indeterminates and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An example of a polynomial of a single indeterminate x is x2 − 4x + 7. An example with three indeterminates is x3 + 2xyz2 − yz + 1.

David Hilbert

David Hilbert

David Hilbert was a German mathematician and one of the most influential mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory, the calculus of variations, commutative algebra, algebraic number theory, the foundations of geometry, spectral theory of operators and its application to integral equations, mathematical physics, and the foundations of mathematics.

Algebraic geometry

Algebraic geometry

Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical problems about these sets of zeros.

Field (mathematics)

Field (mathematics)

In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational and real numbers do. A field is thus a fundamental algebraic structure which is widely used in algebra, number theory, and many other areas of mathematics.

Proof by contradiction

Proof by contradiction

In logic and mathematics, proof by contradiction is a form of proof that establishes the truth or the validity of a proposition, by showing that assuming the proposition to be false leads to a contradiction. Proof by contradiction is also known as indirect proof, proof by assuming the opposite, and reductio ad impossibile. It is an example of the weaker logical refutation reductio ad absurdum.

Mathematical induction

Mathematical induction

Mathematical induction is a method for proving that a statement P(n) is true for every natural number n, that is, that the infinitely many cases P(0), P(1), P(2), P(3), ...  all hold. Informal metaphors help to explain this technique, such as falling dominoes or climbing a ladder:Mathematical induction proves that we can climb as high as we like on a ladder, by proving that we can climb onto the bottom rung and that from each rung we can climb up to the next one.

Algorithm

Algorithm

In mathematics and computer science, an algorithm is a finite sequence of rigorous instructions, typically used to solve a class of specific problems or to perform a computation. Algorithms are used as specifications for performing calculations and data processing. More advanced algorithms can perform automated deductions and use mathematical and logical tests to divert the code execution through various routes. Using human characteristics as descriptors of machines in metaphorical ways was already practiced by Alan Turing with terms such as "memory", "search" and "stimulus".

Ideal (ring theory)

Ideal (ring theory)

In ring theory, a branch of abstract algebra, an ideal of a ring is a special subset of its elements. Ideals generalize certain subsets of the integers, such as the even numbers or the multiples of 3. Addition and subtraction of even numbers preserves evenness, and multiplying an even number by any integer results in an even number; these closure and absorption properties are the defining properties of an ideal. An ideal can be used to construct a quotient ring in a way similar to how, in group theory, a normal subgroup can be used to construct a quotient group.

Gröbner basis

Gröbner basis

In mathematics, and more specifically in computer algebra, computational algebraic geometry, and computational commutative algebra, a Gröbner basis is a particular kind of generating set of an ideal in a polynomial ring K[x1, ..., xn] over a field K. A Gröbner basis allows many important properties of the ideal and the associated algebraic variety to be deduced easily, such as the dimension and the number of zeros when it is finite. Gröbner basis computation is one of the main practical tools for solving systems of polynomial equations and computing the images of algebraic varieties under projections or rational maps.

Proof

Theorem. If is a left (resp. right) Noetherian ring, then the polynomial ring is also a left (resp. right) Noetherian ring.

Remark. We will give two proofs, in both only the "left" case is considered; the proof for the right case is similar.

First proof

Suppose is a non-finitely generated left ideal. Then by recursion (using the axiom of dependent choice) there is a sequence of polynomials such that if is the left ideal generated by then is of minimal degree. It is clear that is a non-decreasing sequence of natural numbers. Let be the leading coefficient of and let be the left ideal in generated by . Since is Noetherian the chain of ideals

must terminate. Thus for some integer . So in particular,

Now consider

whose leading term is equal to that of ; moreover, . However, , which means that has degree less than , contradicting the minimality.

Second proof

Let be a left ideal. Let be the set of leading coefficients of members of . This is obviously a left ideal over , and so is finitely generated by the leading coefficients of finitely many members of ; say . Let be the maximum of the set , and let be the set of leading coefficients of members of , whose degree is . As before, the are left ideals over , and so are finitely generated by the leading coefficients of finitely many members of , say

with degrees . Now let be the left ideal generated by:

We have and claim also . Suppose for the sake of contradiction this is not so. Then let be of minimal degree, and denote its leading coefficient by .

Case 1: . Regardless of this condition, we have , so is a left linear combination
of the coefficients of the . Consider
which has the same leading term as ; moreover while . Therefore and , which contradicts minimality.
Case 2: . Then so is a left linear combination
of the leading coefficients of the . Considering
we yield a similar contradiction as in Case 1.

Thus our claim holds, and which is finitely generated.

Note that the only reason we had to split into two cases was to ensure that the powers of multiplying the factors were non-negative in the constructions.

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Noetherian ring

Noetherian ring

In mathematics, a Noetherian ring is a ring that satisfies the ascending chain condition on left and right ideals; if the chain condition is satisfied only for left ideals or for right ideals, then the ring is said left-Noetherian or right-Noetherian respectively. That is, every increasing sequence of left ideals has a largest element; that is, there exists an n such that:

Polynomial ring

Polynomial ring

In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring formed from the set of polynomials in one or more indeterminates with coefficients in another ring, often a field.

Axiom of dependent choice

Axiom of dependent choice

In mathematics, the axiom of dependent choice, denoted by , is a weak form of the axiom of choice that is still sufficient to develop most of real analysis. It was introduced by Paul Bernays in a 1942 article that explores which set-theoretic axioms are needed to develop analysis.

Degree of a polynomial

Degree of a polynomial

In mathematics, the degree of a polynomial is the highest of the degrees of the polynomial's monomials with non-zero coefficients. The degree of a term is the sum of the exponents of the variables that appear in it, and thus is a non-negative integer. For a univariate polynomial, the degree of the polynomial is simply the highest exponent occurring in the polynomial. The term order has been used as a synonym of degree but, nowadays, may refer to several other concepts.

Natural number

Natural number

In mathematics, the natural numbers are those numbers used for counting and ordering. Numbers used for counting are called cardinal numbers, and numbers used for ordering are called ordinal numbers. Natural numbers are sometimes used as labels, known as nominal numbers, having none of the properties of numbers in a mathematical sense.

Integer

Integer

An integer is the number zero (0), a positive natural number or a negative integer with a minus sign. The negative numbers are the additive inverses of the corresponding positive numbers. In the language of mathematics, the set of integers is often denoted by the boldface Z or blackboard bold .

Applications

Let be a Noetherian commutative ring. Hilbert's basis theorem has some immediate corollaries.

  1. By induction we see that will also be Noetherian.
  2. Since any affine variety over (i.e. a locus-set of a collection of polynomials) may be written as the locus of an ideal and further as the locus of its generators, it follows that every affine variety is the locus of finitely many polynomials — i.e. the intersection of finitely many hypersurfaces.
  3. If is a finitely-generated -algebra, then we know that , where is an ideal. The basis theorem implies that must be finitely generated, say , i.e. is finitely presented.

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Commutative ring

Commutative ring

In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not specific to commutative rings. This distinction results from the high number of fundamental properties of commutative rings that do not extend to noncommutative rings.

Corollary

Corollary

In mathematics and logic, a corollary is a theorem of less importance which can be readily deduced from a previous, more notable statement. A corollary could, for instance, be a proposition which is incidentally proved while proving another proposition; it might also be used more casually to refer to something which naturally or incidentally accompanies something else.

Affine variety

Affine variety

In algebraic geometry, an affine variety, or affine algebraic variety, over an algebraically closed field k is the zero-locus in the affine space kn of some finite family of polynomials of n variables with coefficients in k that generate a prime ideal. If the condition of generating a prime ideal is removed, such a set is called an (affine) algebraic set. A Zariski open subvariety of an affine variety is called a quasi-affine variety.

Intersection (set theory)

Intersection (set theory)

In set theory, the intersection of two sets and denoted by is the set containing all elements of that also belong to or equivalently, all elements of that also belong to

Hypersurface

Hypersurface

In geometry, a hypersurface is a generalization of the concepts of hyperplane, plane curve, and surface. A hypersurface is a manifold or an algebraic variety of dimension n − 1, which is embedded in an ambient space of dimension n, generally a Euclidean space, an affine space or a projective space. Hypersurfaces share, with surfaces in a three-dimensional space, the property of being defined by a single implicit equation, at least locally, and sometimes globally.

Formal proofs

Formal proofs of Hilbert's basis theorem have been verified through the Mizar project (see HILBASIS file) and Lean (see ring_theory.polynomial).

Source: "Hilbert's basis theorem", Wikipedia, Wikimedia Foundation, (2022, November 25th), https://en.wikipedia.org/wiki/Hilbert's_basis_theorem.

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References
  1. ^ Hilbert, David (1890). "Ueber die Theorie der algebraischen Formen". Mathematische Annalen. 36 (4): 473–534. doi:10.1007/BF01208503. ISSN 0025-5831. S2CID 179177713.
Further reading
  • Cox, Little, and O'Shea, Ideals, Varieties, and Algorithms, Springer-Verlag, 1997.

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