Algebraic geometry - Regular functions

Algebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with geometry. It can be seen as the study of solution sets of systems of algebraic equations. When there is more than one variable, geometric considerations enter and are important to understand the phenomenon. One can say that the subject starts where equation solving leaves off, and it becomes at least as important to understand the totality of solutions of a system of equations as to find som ...

Including:

• Algebraic geometry - Zeroes of simultaneous polynomials
• Algebraic geometry - Affine varieties
• Algebraic geometry - Regular functions
• Algebraic geometry - The category of affine varieties
• Algebraic geometry - Projective space
• Algebraic geometry - The modern viewpoint
• Algebraic geometry - Notes and history

Read more here: » Algebraic geometry: Encyclopedia - Algebraic geometry

Just as continuous functions are the natural maps on topological spaces and smooth functions are the natural maps on differentiable manifolds, there is a natural class of functions on an algebraic set, called regular functions. A regular function on an algebraic set V contained in is defined to be the restriction of a regular function on , in the sense we defined above. It may seem unnaturally restrictive to require that a regular function always extend to the ambient space, but it is very similar to the situation in a normal ...

See also:

Algebraic geometry, Algebraic geometry - Zeroes of simultaneous polynomials, Algebraic geometry - Affine varieties, Algebraic geometry - Regular functions, Algebraic geometry - The category of affine varieties, Algebraic geometry - Projective space, Algebraic geometry - The modern viewpoint, Algebraic geometry - Notes and history

Read more here: » Algebraic geometry: Encyclopedia II - Algebraic geometry - Regular functions

First we start with a field k. In classical algebraic geometry, this field was always C, the complex numbers, but many of the same results are true if we assume only that k is algebraically closed. We define , called the affine n-space over k, to be kn. The purpose of this apparently superfluous notation is to emphasize that one `forgets' the vector space structure that kn carries. Abstractly speaking, is, for the moment, just a collection of points. Henceforth we will drop the k in and instea ...

See also:

Algebraic geometry, Algebraic geometry - Zeroes of simultaneous polynomials, Algebraic geometry - Affine varieties, Algebraic geometry - Regular functions, Algebraic geometry - The category of affine varieties, Algebraic geometry - Projective space, Algebraic geometry - The modern viewpoint, Algebraic geometry - Notes and history

Read more here: » Algebraic geometry: Encyclopedia II - Algebraic geometry - Affine varieties

First we start with a field k. In classical algebraic geometry, this field was always , the complex numbers, but many of the same results are true if we assume only that k is algebraically closed. We define , called the affine n-space over k, to be kn. The purpose of this apparently superfluous notation is to emphasize that one `forgets' the vector space structure that kn carries. Abstractly speaking, is, for the moment, just a collection of points. Henceforth we will drop the k in and instea ...

See also:

Algebraic geometry, Algebraic geometry - Zeroes of simultaneous polynomials, Algebraic geometry - Affine varieties, Algebraic geometry - Regular functions, Algebraic geometry - The category of affine varieties, Algebraic geometry - Projective space, Algebraic geometry - The modern viewpoint, Algebraic geometry - Notes and history

Read more here: » Algebraic geometry: Encyclopedia II - Algebraic geometry - Affine varieties

Consider the variety V(y=x2). If we draw it, we get a parabola. As x increases, the slope of the line from the origin to the point (x,x2) becomes larger and larger. As x decreases, the slope of the same line becomes smaller and smaller. Compare this to the variety V(y=x3). This is a cubic equation. As x increases, the slope of the line from the origin to the point (x,x3) becomes larger and larger just as before. But unlike before, as x decreases, the slope of the same line again becomes larg ...

See also:

Algebraic geometry, Algebraic geometry - Zeroes of simultaneous polynomials, Algebraic geometry - Affine varieties, Algebraic geometry - Regular functions, Algebraic geometry - The category of affine varieties, Algebraic geometry - Projective space, Algebraic geometry - The modern viewpoint, Algebraic geometry - Notes and history

Read more here: » Algebraic geometry: Encyclopedia II - Algebraic geometry - Projective space

Using regular functions from an affine variety to , we can define regular functions from one affine variety to another. First we will define a regular function from a variety into affine space: Let V be a variety contained in . Choose m regular functions on V, and call them f1,...,fm. We define a regular function f from V to by letting f(t1,...,tn)=(f1,...,fm). In other words, each fi determines one coordinate of the range of f. If V' is a variety contained in , we say that f is a ...

See also:

Algebraic geometry, Algebraic geometry - Zeroes of simultaneous polynomials, Algebraic geometry - Affine varieties, Algebraic geometry - Regular functions, Algebraic geometry - The category of affine varieties, Algebraic geometry - Projective space, Algebraic geometry - The modern viewpoint, Algebraic geometry - Notes and history

Read more here: » Algebraic geometry: Encyclopedia II - Algebraic geometry - The category of affine varieties

In classical algebraic geometry, the main objects of interest are the vanishing sets of collections of polynomials, meaning the set of all points that simultaneously satisfy one or more polynomial equations. For instance, the two-dimensional sphere in three-dimensional Euclidean space could be defined as the set of all points (x,y,z) with x2See also:

Algebraic geometry, Algebraic geometry - Zeroes of simultaneous polynomials, Algebraic geometry - Affine varieties, Algebraic geometry - Regular functions, Algebraic geometry - The category of affine varieties, Algebraic geometry - Projective space, Algebraic geometry - The modern viewpoint, Algebraic geometry - Notes and history

Read more here: » Algebraic geometry: Encyclopedia II - Algebraic geometry - Zeroes of simultaneous polynomials

Algebraic geometry was developed largely by the Italian geometers in the early part of the 20th century. Enriques classified algebraic surfaces up to birational isomorphism. The style of the Italian school was very intuitive and does not meet the modern standards of rigor. By the 1930s and 1940s, Oscar Zariski, André Weil and others realized that algebraic geometry needed to be rebuilt on foundations of commutative algebra and valuation theory. Commutative algebra (earlier known as elimination theory and then ideal theory, and refoun ...

See also:

Algebraic geometry, Algebraic geometry - Zeroes of simultaneous polynomials, Algebraic geometry - Affine varieties, Algebraic geometry - Regular functions, Algebraic geometry - The category of affine varieties, Algebraic geometry - Projective space, Algebraic geometry - The modern viewpoint, Algebraic geometry - Notes and history

Read more here: » Algebraic geometry: Encyclopedia II - Algebraic geometry - Notes and history