Concept

Affine variety

In algebraic geometry, an affine algebraic set is the set of the common zeros over an algebraically closed field k of some family of polynomials in the polynomial ring An affine variety or affine algebraic variety, is an affine algebraic set such that the ideal generated by the defining polynomials is prime. Some texts call variety any algebraic set, and irreducible variety an algebraic set whose defining ideal is prime (affine variety in the above sense). In some contexts (see, for example, Hilbert's Nullstellensatz), it is useful to distinguish the field k in which the coefficients are considered, from the algebraically closed field K (containing k) over which the common zeros are considered (that is, the points of the affine algebric set are in Kn). In this case, the variety is said defined over k, and the points of the variety that belong to kn are said k-rational or rational over k. In the common case where k is the field of real numbers, a k-rational point is called a real point. When the field k is not specified, a rational point is a point that is rational over the rational numbers. For example, Fermat's Last Theorem asserts that the affine algebraic variety (it is a curve) defined by xn + yn − 1 = 0 has no rational points for any integer n greater than two. An affine algebraic set is the set of solutions in an algebraically closed field k of a system of polynomial equations with coefficients in k. More precisely, if are polynomials with coefficients in k, they define an affine algebraic set An affine (algebraic) variety is an affine algebraic set which is not the union of two proper affine algebraic subsets. Such an affine algebraic set is often said to be irreducible. If X is an affine algebraic set, and I is the ideal of all polynomials that are zero on X, then the quotient ring is called the of X. If X is an affine variety, then I is prime, so the coordinate ring is an integral domain. The elements of the coordinate ring R are also called the regular functions or the polynomial functions on the variety.

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Official source

https://en.wikipedia.org/wiki/Affine_variety

About this result

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The set of real matrices of upper-bounded rank is a real algebraic variety called the real generic determinantal variety. An explicit description of the tangent cone to that variety is given in Theorem 3.2 of Schneider and Uschmajew (SIAM J. Optim. 25(1):622–646, 2015). The present paper shows that the proof therein is incomplete and provides a proof. It also reviews equivalent descriptions of the tangent cone to that variety. Moreover, it shows that the tangent cone and the algebraic tangent cone to that v

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Let 𝐴 ∈ℤ𝑚×𝑛 be an integer matrix with entries bounded by Δ in absolute value. Cook et al. (1986) have shown that there exists a universal matrix 𝐵 ∈ℤ𝑚′×𝑛 with the following property: For each 𝑏 ∈ℤ𝑚, there exists a 𝑡 ∈ℤ𝑚′ such that the integer hull of the polyhedron 𝑃 ={𝑥 ∈ℝ𝑛 :𝐴⁢𝑥 ≤𝑏} is described by 𝑃𝐼 ={𝑥 ∈ℝ𝑛 :𝐵⁢𝑥 ≤𝑡}. Our main result is that 𝑡 is an affine function of 𝑏 as long as 𝑏 is from a fixed equivalence class of the lattice 𝐷 ⋅ℤ𝑚. Here 𝐷 ∈ℕ is a number that depends on 𝑛 and Δ only. Furthermore, 𝐷 a

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In commutative algebra and algebraic geometry, localization is a formal way to introduce the "denominators" to a given ring or module. That is, it introduces a new ring/module out of an existing ring/module R, so that it consists of fractions such that the denominator s belongs to a given subset S of R. If S is the set of the non-zero elements of an integral domain, then the localization is the field of fractions: this case generalizes the construction of the field of rational numbers from the ring of integers.

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