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Any algebraic set is a finite union of irreducible algebraic sets and this decomposition is unique. Thus its elements are called the irreducible components of the algebraic set. An irreducible algebraic set is also called a variety.

It turns out that an algebraic set is a variety if and only if it may be defined as the vanishing set of a prime ideal of the polynomial ring. Some authors do not make a clear distinction between algebraic sets and varieties and use irreducible variety to make the distinction when needed. 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 or polynomial functions.

A regular function on an algebraic set V contained in A n is the restriction to V of a regular function on A n. For an algebraic set defined on the field of the complex numbers, the regular functions are smooth and even analytic. 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 topological space , where the Tietze extension theorem guarantees that a continuous function on a closed subset always extends to the ambient topological space.

Just as with the regular functions on affine space, the regular functions on V form a ring, which we denote by k [ V ]. This ring is called the coordinate ring of V. Since regular functions on V come from regular functions on A n , there is a relationship between the coordinate rings.

Using regular functions from an affine variety to A 1 , we can define regular maps from one affine variety to another. First we will define a regular map from a variety into affine space: Let V be a variety contained in A n. Choose m regular functions on V , and call them f 1 , In other words, each f i determines one coordinate of the range of f. The definition of the regular maps apply also to algebraic sets. The regular maps are also called morphisms , as they make the collection of all affine algebraic sets into a category , where the objects are the affine algebraic sets and the morphisms are the regular maps.

The affine varieties is a subcategory of the category of the algebraic sets. This defines an equivalence of categories between the category of algebraic sets and the opposite category of the finitely generated reduced k -algebras. This equivalence is one of the starting points of scheme theory.

In contrast to the preceding sections, this section concerns only varieties and not algebraic sets. On the other hand, the definitions extend naturally to projective varieties next section , as an affine variety and its projective completion have the same field of functions.

If V is an affine variety, its coordinate ring is an integral domain and has thus a field of fractions which is denoted k V and called the field of the rational functions on V or, shortly, the function field of V. Its elements are the restrictions to V of the rational functions over the affine space containing V. The domain of a rational function f is not V but the complement of the subvariety a hypersurface where the denominator of f vanishes.

As with regular maps, one may define a rational map from a variety V to a variety V '. As with the regular maps, the rational maps from V to V ' may be identified to the field homomorphisms from k V ' to k V.

Two affine varieties are birationally equivalent if there are two rational functions between them which are inverse one to the other in the regions where both are defined. Equivalently, they are birationally equivalent if their function fields are isomorphic. An affine variety is a rational variety if it is birationally equivalent to an affine space.

This means that the variety admits a rational parameterization. The problem of resolution of singularities is to know if every algebraic variety is birationally equivalent to a variety whose projective completion is nonsingular see also smooth completion. It was solved in the affirmative in characteristic 0 by Heisuke Hironaka in and is yet unsolved in finite characteristic. Just as the formulas for the roots of second, third, and fourth degree polynomials suggest extending real numbers to the more algebraically complete setting of the complex numbers, many properties of algebraic varieties suggest extending affine space to a more geometrically complete projective space.

If we draw it, we get a parabola. As x goes to negative infinity, the slope of the same line goes to negative infinity. This is a cubic curve. But unlike before, as x goes to negative infinity, the slope of the same line goes to positive infinity as well; the exact opposite of the parabola. The consideration of the projective completion of the two curves, which is their prolongation "at infinity" in the projective plane , allows us to quantify this difference: Also, both curves are rational, as they are parameterized by x , and the Riemann-Roch theorem implies that the cubic curve must have a singularity, which must be at infinity, as all its points in the affine space are regular.

Thus many of the properties of algebraic varieties, including birational equivalence and all the topological properties, depend on the behavior "at infinity" and so it is natural to study the varieties in projective space.

Furthermore, the introduction of projective techniques made many theorems in algebraic geometry simpler and sharper: For these reasons, projective space plays a fundamental role in algebraic geometry. In this case, one says that the polynomial vanishes at the corresponding point of P n. This allows us to define a projective algebraic set in P n as the set V f 1 , Like for affine algebraic sets, there is a bijection between the projective algebraic sets and the reduced homogeneous ideals which define them.

The projective varieties are the projective algebraic sets whose defining ideal is prime. Every projective algebraic set may be uniquely decomposed into a finite union of projective varieties. The only regular functions which may be defined properly on a projective variety are the constant functions. Thus this notion is not used in projective situations.

On the other hand, the field of the rational functions or function field is a useful notion, which, similarly to the affine case, is defined as the set of the quotients of two homogeneous elements of the same degree in the homogeneous coordinate ring.

The fact that the field of the real numbers is an ordered field cannot be ignored in such a study. It follows that real algebraic geometry is not only the study of the real algebraic varieties, but has been generalized to the study of the semi-algebraic sets , which are the solutions of systems of polynomial equations and polynomial inequalities. One of the challenging problems of real algebraic geometry is the unsolved Hilbert's sixteenth problem: Decide which respective positions are possible for the ovals of a nonsingular plane curve of degree 8.

Since then, most results in this area are related to one or several of these items either by using or improving one of these algorithms, or by finding algorithms whose complexity is simply exponential in the number of the variables. A body of mathematical theory complementary to symbolic methods called numerical algebraic geometry has been developed over the last several decades.

The main computational method is homotopy continuation. This supports, for example, a model of floating point computation for solving problems of algebraic geometry. In fact they may contain, in the worst case, polynomials whose degree is doubly exponential in the number of variables and a number of polynomials which is also doubly exponential.

However, this is only a worst case complexity, and the complexity bound of Lazard's algorithm of may frequently apply. It follows that the best implementations allow one to compute almost routinely with algebraic sets of degree more than CAD is an algorithm which was introduced in by G. Collins to implement with an acceptable complexity the Tarski—Seidenberg theorem on quantifier elimination over the real numbers.

This theorem concerns the formulas of the first-order logic whose atomic formulas are polynomial equalities or inequalities between polynomials with real coefficients. The complexity of CAD is doubly exponential in the number of variables. This means that CAD allows, in theory, to solve every problem of real algebraic geometry which may be expressed by such a formula, that is almost every problem concerning explicitly given varieties and semi-algebraic sets.

This implies that, unless if most polynomials appearing in the input are linear, it may not solve problems with more than four variables. Since , most of the research on this subject is devoted either to improve CAD or to find alternate algorithms in special cases of general interest. As an example of the state of art, there are efficient algorithms to find at least a point in every connected component of a semi-algebraic set, and thus to test if a semi-algebraic set is empty.

On the other hand, CAD is yet, in practice, the best algorithm to count the number of connected components. The basic general algorithms of computational geometry have a double exponential worst case complexity. During the last 20 years of 20th century, various algorithms have been introduced to solve specific subproblems with a better complexity.

The main algorithms of real algebraic geometry which solve a problem solved by CAD are related to the topology of semi-algebraic sets. One may cite counting the number of connected components , testing if two points are in the same components or computing a Whitney stratification of a real algebraic set.

Therefore, these algorithms have never been implemented and this is an active research area to search for algorithms with have together a good asymptotic complexity and a good practical efficiency. The modern approaches to algebraic geometry redefine and effectively extend the range of basic objects in various levels of generality to schemes, formal schemes , ind-schemes , algebraic spaces , algebraic stacks and so on.

The need for this arises already from the useful ideas within theory of varieties, e. Most remarkably, in late s, algebraic varieties were subsumed into Alexander Grothendieck 's concept of a scheme. Their local objects are affine schemes or prime spectra which are locally ringed spaces which form a category which is antiequivalent to the category of commutative unital rings, extending the duality between the category of affine algebraic varieties over a field k , and the category of finitely generated reduced k -algebras.

The gluing is along Zariski topology; one can glue within the category of locally ringed spaces, but also, using the Yoneda embedding, within the more abstract category of presheaves of sets over the category of affine schemes. The Zariski topology in the set theoretic sense is then replaced by a Grothendieck topology. Sheaves can be furthermore generalized to stacks in the sense of Grothendieck, usually with some additional representability conditions leading to Artin stacks and, even finer, Deligne-Mumford stacks , both often called algebraic stacks.

Sometimes other algebraic sites replace the category of affine schemes. For example, Nikolai Durov has introduced commutative algebraic monads as a generalization of local objects in a generalized algebraic geometry. Versions of a tropical geometry , of an absolute geometry over a field of one element and an algebraic analogue of Arakelov's geometry were realized in this setup. Another formal generalization is possible to universal algebraic geometry in which every variety of algebras has its own algebraic geometry.

The term variety of algebras should not be confused with algebraic variety. The language of schemes, stacks and generalizations has proved to be a valuable way of dealing with geometric concepts and became cornerstones of modern algebraic geometry.

Algebraic stacks can be further generalized and for many practical questions like deformation theory and intersection theory, this is often the most natural approach. One can extend the Grothendieck site of affine schemes to a higher categorical site of derived affine schemes , by replacing the commutative rings with an infinity category of differential graded commutative algebras , or of simplicial commutative rings or a similar category with an appropriate variant of a Grothendieck topology.

One can also replace presheaves of sets by presheaves of simplicial sets or of infinity groupoids. Then, in presence of an appropriate homotopic machinery one can develop a notion of derived stack as such a presheaf on the infinity category of derived affine schemes, which is satisfying certain infinite categorical version of a sheaf axiom and to be algebraic, inductively a sequence of representability conditions.

Another noncommutative version of derived algebraic geometry, using A-infinity categories has been developed from early s by Maxim Kontsevich and followers. Some of the roots of algebraic geometry date back to the work of the Hellenistic Greeks from the 5th century BC. The Delian problem , for instance, was to construct a length x so that the cube of side x contained the same volume as the rectangular box a 2 b for given sides a and b.

This was done, for instance, by Ibn al-Haytham in the 10th century AD. The geometrical approach to construction problems, rather than the algebraic one, was favored by most 16th and 17th century mathematicians, notably Blaise Pascal who argued against the use of algebraic and analytical methods in geometry.

They were interested primarily in the properties of algebraic curves , such as those defined by Diophantine equations in the case of Fermat , and the algebraic reformulation of the classical Greek works on conics and cubics in the case of Descartes. Pascal and Desargues also studied curves, but from the purely geometrical point of view: Ultimately, the analytic geometry of Descartes and Fermat won out, for it supplied the 18th century mathematicians with concrete quantitative tools needed to study physical problems using the new calculus of Newton and Leibniz.

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