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...k |author-link=Igor Shafarevich |first=I.R. |last=Shafarevich |title=Basic Algebraic Geometry |url=https://books.google.com/books?id=m6nwCAAAQBAJ |date=2012 |pu Multi-homogeneous Bézout theorem provides such a better bound when the unknowns may be split ...

...o <math>\mathbb{R}^{2n}</math> for some ''n'', but is not isomorphic as an algebraic variety to <math>\mathbb{C}^n</math>.<ref>{{citation | contribution = The role of exotic affine spaces in the classification of homogeneous affine varieties ...

...tract [[algebraic geometry]] and to describe some basic uses of projective spaces. == Homogeneous polynomial ideals== ...

...</math> of structures; the elements of <math>X_i</math> are said to be "'''homogeneous''' of '''degree''' ''i''{{-"}}. ** An algebraic structure is said to be [[doubly graded]] if the index set is a direct prod ...

...This vector field is radial in the sense that it vanishes uniformly on 0-homogeneous functions, that is, the functions that are invariant by homothetic rescalin ...o a 0-homogeneous function on ''V'' (again partially defined). We obtain 1-homogeneous vector fields by multiplying the Euler vector field by such functions. This ...

...math> is called convex if the pullback of the tangent bundle to a stable [[Algebraic curve|rational curve]] <math>f:C \to X</math> has globally generated sectio There are many examples of convex spaces, including the following. ...

The fibres are therefore vector spaces, and the projection ''p'' is a [[vector bundle]] over the [[Grassmannian]], As a [[homogeneous space]], the affine Grassmannian of an ''n''-dimensional vector space ''V'' ...

Since real oriented Grassmannians can be expressed as a [[homogeneous space]] by: == Simplest classifying spaces == ...

...[[Convex function|convex]] [[Order theory|order theoretic]] views on state spaces of physical systems. .... The cone is called ''self-dual'' when <math>C=C^*</math>. It is called ''homogeneous'' when to any two points <math>a,b \in C</math> there is a real [[linear tr ...

for [[homogeneous element]]s ''x'', ''y'' in ''M'' of degree |''x''&hairsp;|, |''y''&hairsp;| * [[David Eisenbud]], ''Commutative Algebra. With a view toward algebraic geometry'', [[Graduate Texts in Mathematics]], vol 150, [[Springer-Verlag]] ...

In the [[mathematics|mathematical]] fields of [[Lie theory]] and [[algebraic topology]], the notion of '''Cartan pair''' is a technical condition on the ...of [[symmetric algebra]]s is induced by the restriction map of dual vector spaces <math>\mathfrak{g}^* \to \mathfrak{k}^*</math>. ...

Since real oriented Grassmannians can be expressed as a [[homogeneous space]] by: == Simplest classifying spaces == ...

...bra)|rank]]s. Their significance comes from the fact that many examples in algebraic geometry are of this form, such as the [[Segre embedding]] of a product of ...ls is a '''determinantal ideal'''. Since the equations defining minors are homogeneous, one can consider ''Y''_&nbsp;''r'' either as an [[affine variety ...

...m depends neither on <math>X</math> nor on <math>p</math>. All two-point [[homogeneous space]]s are globally Osserman, including [[Euclidean space]]s <math>\mathb ...fundamental in studying Osserman manifolds. An [[Riemann curvature tensor|algebraic curvature tensor]] <math>R</math> in <math>\mathbb{R}^n</math> has a <math> ...

...be defined as a [[projective space]] over a finite field.<ref>"Projective spaces over a finite field, otherwise known as Galois geometries, ...", {{Harv|Hir Objects of study include [[affine space|affine]] and projective spaces over finite fields and various structures that are contained in them. In pa ...

...=Bass, H.|year=1973|title=Some problems in 'classical' algebraic K-theory. Algebraic K-Theory II|publisher=Springer-Verlag|location=Berlin-Heidelberg-New York}} ...ility results in A^1-homotopy theory II: principal bundles and homogeneous spaces|author=Asok|first1=Aravind|last2=Hoyois|first2=Marc|last3=Wendt|first3=Matt ...

{{about|quadrics in [[algebraic geometry]]|quadrics over the [[real number]]s|quadric}} ...er a [[field (mathematics)|field]]. Quadrics are fundamental examples in [[algebraic geometry]]. The theory is simplified by working in [[projective space]] rat ...

...ebraically closed field, embedded in a [[algebraic geometry of projective spaces|projective space]] <math>\mathbf P^n</math>. ...: for a connected [[algebraic group]] ''G'', and any [[homogeneous variety|homogeneous ''G''-variety]] ''X'', and two varieties ''Y'' and ''Z'' mapping to ''X'', ...

...>t^n</math> gives the dimension (or rank) of the sub-structure of elements homogeneous of degree <math>n</math>. It is closely related to the [[Hilbert polynomia ...le over any [[commutative ring]] ''R'' in which each submodule of elements homogeneous of a fixed degree ''n'' is [[Free module|free]] of finite rank; it suffices ...

...[[polynomial]]s. It is also called a '''regular map'''. A morphism from an algebraic variety to the [[affine line]] is also called a '''regular function'''. ...called '''biregular''', and the biregular maps are the [[isomorphism]]s of algebraic varieties. Because regular and biregular are very restrictive conditions – ...