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In mathematics, a **solution set** is the set of values that satisfy a given set of equations or inequalities.

For example, for a set of polynomials over a ring , the solution set is the subset of on which the polynomials all vanish (evaluate to 0), formally

The feasible region of a constrained optimization problem is the solution set of the constraints.

- The solution set of the single equation is the set {0}.
- For any non-zero polynomial over the complex numbers in one variable, the solution set is made up of finitely many points.
- However, for a complex polynomial in more than one variable the solution set has no isolated points.

In algebraic geometry, solution sets are called algebraic sets if there are no inequalities. Over the reals, and with inequalities, there are called semialgebraic sets.

More generally, the **solution set** to an arbitrary collection *E* of relations (*E _{i}*) (

(Instead of relations depending on unknowns, one should speak more correctly of predicates, the collection *E* is their logical conjunction, and the solution set is the inverse image of the boolean value *true* by the associated boolean-valued function.)

The above meaning is a special case of this one, if the set of polynomials *f _{i}* if interpreted as the set of equations

- The solution set for
*E*= {*x*+*y*= 0 } with respect to is*S*= { (*a*,−*a*) :*a*∈**R**}. - The solution set for
*E*= {*x*+*y*= 0 } with respect to is*S*= { −*y*}. (Here,*y*is not "declared" as an unknown, and thus to be seen as a parameter on which the equation, and therefore the solution set, depends.) - The solution set for with respect to is the interval
*S*= [0,2] (since is undefined for negative values of*x*). - The solution set for with respect to is
*S*= 2π**Z**(see Euler's identity).

In mathematics, the **absolute value** or **modulus** of a real number x, denoted |*x*|, is the non-negative value of x without regard to its sign. Namely, |*x*| = *x* if x is positive, and |*x*| = −*x* if x is negative, and |0| = 0. For example, the absolute value of 3 is 3, and the absolute value of −3 is also 3. The absolute value of a number may be thought of as its distance from zero.

In mathematics, a **complex number** is a number that can be expressed in the form *a* + *bi*, where a and b are real numbers, and i is a symbol called the imaginary unit, and satisfying the equation *i*^{2} = -1. Because no "real" number satisfies this equation, i was called an imaginary number by René Descartes. For the complex number *a* + *bi*, a is called the **real part** and b is called the **imaginary part**. The set of complex numbers is denoted by either of the symbols or **C**. Despite the historical nomenclature "imaginary", complex numbers are regarded in the mathematical sciences as just as "real" as the real numbers and are fundamental in many aspects of the scientific description of the natural world.

In mathematics, an **equation** is a statement that asserts the equality of two expressions, which are connected by the equals sign "=". The word *equation* and its cognates in other languages may have subtly different meanings; for example, in French an *équation* is defined as containing one or more variables, while in English, any equality is an equation.

**Hilbert's Nullstellensatz** is a theorem that establishes a fundamental relationship between geometry and algebra. This relationship is the basis of algebraic geometry, a branch of mathematics. It relates algebraic sets to ideals in polynomial rings over algebraically closed fields. This relationship was discovered by David Hilbert who proved the Nullstellensatz and several other important related theorems named after him.

In mathematics and physical science, **spherical harmonics** are special functions defined on the surface of a sphere. They are often employed in solving partial differential equations in many scientific fields.

In mathematics, an **implicit equation** is a relation of the form *R*(*x*_{1},…, *x _{n}*) = 0, where R is a function of several variables. For example, the implicit equation of the unit circle is

In mathematics, a **zero** of a real-, complex-, or generally vector-valued function , is a member of the domain of such that *vanishes* at ; that is, the function attains the value of 0 at , or equivalently, is the solution to the equation . A "zero" of a function is thus an input value that produces an output of 0.

In mathematics, and more specifically in computer algebra, computational algebraic geometry, and computational commutative algebra, a **Gröbner basis** is a particular kind of generating set of an ideal in a polynomial ring *K*[*x*_{1}, …, *x*_{n}] over a field *K*. A Gröbner basis allows many important properties of the ideal and the associated algebraic variety to be deduced easily, such as the dimension and the number of zeros when it is finite. Gröbner basis computation is one of the main practical tools for solving systems of polynomial equations and computing the images of algebraic varieties under projections or rational maps.

In mathematics, to **solve an equation** is to find its **solutions**, which are the values that fulfill the condition stated by the equation, consisting generally of two expressions related by an equals sign. When seeking a solution, one or more variables are designated as *unknowns*. A solution is an assignment of values to the unknown variables that makes the equality in the equation true. In other words, a solution is a value or a collection of values such that, when substituted for the unknowns, the equation becomes an equality. A solution of an equation is often called a **root** of the equation, particularly but not only for polynomial equations. The set of all solutions of an equation is its solution set.

In mathematics, an **algebraic function** is a function that can be defined as the root of a polynomial equation. Quite often algebraic functions are algebraic expressions using a finite number of terms, involving only the algebraic operations addition, subtraction, multiplication, division, and raising to a fractional power. Examples of such functions are:

In mathematics, a **differential-algebraic system of equations** (**DAEs**) is a system of equations that either contains differential equations and algebraic equations, or is equivalent to such a system. Such systems occur as the general form of differential equations for vector–valued functions *x* in one independent variable *t*,

**Hidden Fields Equations (HFE)**, also known as **HFE trapdoor function**, is a public key cryptosystem which was introduced at Eurocrypt in 1996 and proposed by (in French) Jacques Patarin following the idea of the Matsumoto and Imai system. It is based on polynomials over finite fields of different size to disguise the relationship between the private key and public key. HFE is in fact a family which consists of basic HFE and combinatorial versions of HFE. The HFE family of cryptosystems is based on the hardness of the problem of finding solutions to a system of multivariate quadratic equations since it uses private affine transformations to hide the extension field and the private polynomials. Hidden Field Equations also have been used to construct digital signature schemes, e.g. Quartz and Sflash.

In mathematics, a **surface** is a generalization of a plane, which is not necessarily flat – that is, the curvature is not necessarily zero. This is analogous to a curve generalizing a straight line. There are many more precise definitions, depending on the context and the mathematical tools that are used to analyze the surface.

**Difference algebra** is a branch of mathematics concerned with the study of difference equations from the algebraic point of view. Difference algebra is analogous to differential algebra but concerned with difference equations rather than differential equations. As an independent subject it was initiated by Joseph Ritt and his student Richard Cohn.

A **system of polynomial equations** is a set of simultaneous equations *f*_{1} = 0, ..., *f*_{h} = 0 where the *f*_{i} are polynomials in several variables, say *x*_{1}, ..., *x*_{n}, over some field *k*.

In computer algebra, a **triangular decomposition** of a polynomial system S is a set of simpler polynomial systems *S*_{1}, ..., *S _{e}* such that a point is a solution of S if and only if it is a solution of one of the systems

In algebraic geometry, the **main theorem of elimination theory** states that every projective scheme is proper. A version of this theorem predates the existence of scheme theory. It can be stated, proved, and applied in the following more classical setting. Let *k* be a field, denote by the *n*-dimensional projective space over *k*. The main theorem of elimination theory is the statement that for any *n* and any algebraic variety V defined over *k*, the projection map sends Zariski-closed subsets to Zariski-closed subsets.

In algebra and algebraic geometry, the **multi-homogeneous Bézout theorem** is a generalization to multi-homogeneous polynomials of Bézout's theorem, which counts the number of isolated common zeros of a set of homogeneous polynomials. This generalization is due to Igor Shafarevich.

In mathematics a **P-recursive equation** can be solved for **polynomial solutions**. Sergei A. Abramov in 1989 and Marko Petkovšek in 1992 described an algorithm which finds all polynomial solutions of those recurrence equations with polynomial coefficients. The algorithm computes a *degree bound* for the solution in a first step. In a second step an ansatz for a polynomial of this degree is used and the unknown coefficients are computed by a system of linear equations. This article describes this algorithm.

In mathematics a **P-recursive equation** is a linear equation of sequences where the coefficient sequences can be represented as polynomials. P-recursive equations are **linear recurrence equations****with polynomial coefficients**. These equations play an important role in different areas of mathematics, specifically in combinatorics. The sequences which are solutions of these equations are called holonomic, P-recursive or D-finite.

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