Let V be an algebraic set defined as the set of the common zeros of an ideal I in a polynomial ring over a field K, and let A = R / I be the algebra of the polynomials over V. Then the dimension of V is:

For the function field even to be defined, V here must be an irreducible algebraic set ; in which case the function field ( for an affine variety ) is just the field of fractions of the coordinate ring of V. Using polynomial equations, it is easy to define sets that have ' mixed dimension ': a union of a curve and a plane in space, for example.

In the particular case of an algebraic curve C, that is, dimension 1, it follows that any two non-constant functions F and G on C satisfy a polynomial equation P ( F, G )

In variable dimension, iterative methods using the Graver basis of the matrix defining the system, enable the solution of broad classes of linear and nonlinear integer programming problems in polynomial time,

If the dimension, n, is odd, there will be a " dangling " eigenvalue of 1 ; and for any dimension the rest of the polynomial factors into quadratic terms like the one here ( with the two special cases noted ).

It turns out that a group with polynomial growth of order d has isoperimetric dimension d. This holds both for the case of Lie groups and for the Cayley graph of a finitely generated group.

* the Hilbert polynomial of the finitely generated graded ring has degree the dimension of X

For n ≥ 0, this space has dimension n + 1 and forms an irreducible representation under the standard action of G on the polynomial algebra C. Weight vectors are given by monomials

This proof method allows for the efficient construction of a Radon point, in an amount of time that is polynomial in the dimension, by using Gaussian elimination or other efficient algorithms to solve the system of equations for the multipliers.

A randomized algorithm that repeatedly replaces sets of d + 2 points by their Radon point can be used to compute an approximation to a centerpoint of any point set, in an amount of time that is polynomial in both the number of points and the dimension.

The last expression shows that for fixed source dimension n, the target dimension is a polynomial in d of degree n and leading coefficient

They guarantee that the number of iterations of the algorithm is bounded by a polynomial in the dimension and accuracy of the solution.

Let A = K be the ring of polynomials in n variables over a field K. Then the global dimension of A is equal to n. This statement goes back to David Hilbert's foundational work on homological properties of polynomial rings, see Hilbert's syzygy theorem.

Let be a non-constant polynomial function of complex variables such that, so that the set of all complex-vectors with is a complex hypersurface of complex dimension containing the origin of complex-space.

In commutative algebra, the Krull dimension of a ring R, named after Wolfgang Krull ( 1899 – 1971 ), is the supremum of the number of strict inclusions in a chain of prime ideals.

The Krull dimension need not be finite even for a Noetherian ring.

Nagata gave an example of a ring that has infinite Krull dimension even though every prime ideal has finite height.

It follows readily from the definition of the spectrum of a ring, the space of prime ideals of equipped with the Zariski topology, that the Krull dimension of is precisely equal to the irreducible dimension of its spectrum.

In general, if R is a Noetherian ring of dimension d, then the dimension of R is d + 1.

* The ring of integers has dimension 1.

In general, a Noetherian ring is Artinian if and only if its Krull dimension is 0.

If R is a commutative ring, and M is an R-module, we define the Krull dimension of M to be the Krull dimension of the quotient of R making M a faithful module.

The ring of algebraic integers in a number field K is Noetherian, integrally closed, and of dimension one ( to see the last property, observe that for any nonzero ideal I of R, R / I is finite and recall that a finite integral domain is a field ), so by ( DD4 ) R is a Dedekind domain.

Indeed, this is essentially an algebraic translation of these geometric terms: the coordinate ring of any affine variety is, by definition, a finitely generated k-algebra, so Noetherian ; moreover curve means dimension one and nonsingular implies ( and, in dimension one, is equivalent to ) normal, which by definition means integrally closed.

In the other case, the packet is fed to a receive queue and may be transferred to a ring in another dimension.

* Let k be the coordinate ring of the variety V. Then the dimension of V is the transcendence degree of the field of fractions of k over k.

If R is a field, the cycle space is a vector space over R with dimension m-n + c, where c is the number of connected components of G. If R is any commutative ring, the cycle space is a free R-module with rank m-n + c.

For a division ring D construct an n + 1 dimensional vector space over D ( vector space dimension is the number of elements in a basis ).

For finite projective spaces of geometric dimension at least three, Wedderburn's theorem implies that the division ring over which the projective space is defined must be a finite field, GF ( q ), whose order ( that is, number of elements ) is q ( a prime power ).

* Regular ring, a ring such that all its localizations have Krull dimension equal to the minimal number of generators of the maximal ideal

Desargues's theorem is true for the real projective plane, for any projective space defined arithmetically from a field or division ring, for any projective space of dimension unequal to two, and for any projective space in which Pappus's theorem holds.

Besides the lack of an adequate ethical dimension to the Governor's case, one can ask seriously whether our lead over the Russians in quality and quantity of nuclear weapons is so slight as to make the tests absolutely necessary.

It is true for odd numbers of space dimension, where for one dimension the integration is performed over the boundary of an interval with respect to the Dirac measure.

The dimension of the vector space V over the field F can be written as dim < sub > F </ sub >( V ) or as: F, read " dimension of V over F ".

The university is also a member of the Socrates-Erasmus programme for exchange students from all over Europe, promoting advanced engineering and a European dimension.

The trade name Rochlitz Porphyr is the traditional designation for a dimension stone of Saxony with an architectural history over 1, 000 years in Germany.

The dual space of a finite-dimensional vector space is again a finite-dimensional vector space of the same dimension, and these are thus isomorphic, since dimension is the only invariant of finite-dimensional vector spaces over a given field.

The quantity may also be given as a ratio of two different units that have the same dimension ( for instance, light years over meters ).

The real line is a vector space over the field of real numbers ( that is, over itself ) of dimension.

The general linear group GL ( n, R ) over the field of real numbers is a real Lie group of dimension n < sup > 2 </ sup >.

The general linear GL ( n, C ) over the field of complex numbers is a complex Lie group of complex dimension n < sup > 2 </ sup >.

More generally, one can count points of Grassmannian over F: in other words the number of subspaces of a given dimension k. This requires only finding the order of the stabilizer subgroup of one such subspace and dividing into the formula just given, by the orbit-stabilizer theorem.

Crashing half through the target wall, Lizardo had been briefly trapped in the 8th dimension where his mind was taken over by Lord John Whorfin.

They intend to rescue the remaining 8th dimension exiles and take over Planet 10.

If we take the modules to be over a field, then the dimension of the n-th homology of X turns out to be the number of " holes " in X at dimension n.

The sum runs over all nearest neighbor positions on a lattice, of any dimension.

This dimension stone has been in use in architecture for over 1, 000 years.

The extension of European control over Africa and Asia added a further dimension to the rivalry and mutual suspicion which characterized international diplomacy in the decades preceding World War I. France's seizure of Tunisia ( 1881 ) initiated fifteen years of tension with Italy, which had hoped to take the country and which retaliated by allying with Germany and waging a decade-long tariff war with France.