Search Results - Differential. Conrad Wolfram: Teaching kids real math with computers. Alpha Examples - Mathematics. Set theory. Alpha Examples - Mathematics. 2 * X + 5=0. 2 * X^2 + 5=0. Wolfram MathWorld: The Web's Most Extensive Mathematics Resource. Polynomial. A polynomial is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients.
A polynomial in one variable (i.e., a univariate polynomial) with constant coefficients is given by The individual summands with the coefficients (usually) included are called monomials (Becker and Weispfenning 1993, p. 191), whereas the products of the form in the multivariate case, i.e., with the coefficients omitted, are called terms (Becker and Weispfenning 1993, p. 188). However, the term "monomial" is sometimes also used to mean polynomial summands without their coefficients, and in some older works, the definitions of monomial and term are reversed. Care is therefore needed in attempting to distinguish these conflicting usages. The highest power in a univariate polynomial is called its order, or sometimes its degree. Any polynomial with can be expressed as where the product runs over the roots of and it is understood that multiple roots are counted with multiplicity.
Algebraic Geometry. Algebraic geometry is the study of geometries that come from algebra, in particular, from rings.
In classical algebraic geometry, the algebra is the ring of polynomials, and the geometry is the set of zeros of polynomials, called an algebraic variety. For instance, the unit circle is the set of zeros of and is an algebraic variety, as are all of the conic sections. In the twentieth century, it was discovered that the basic ideas of classical algebraic geometry can be applied to any commutative ring with a unit, such as the integers.
The geometry of such a ring is determined by its algebraic structure, in particular its prime ideals. As a consequence, algebraic geometry became very useful in other areas of mathematics, most notably in algebraic number theory. In the latter part of the twentieth century, researchers have tried to extend the relationship between algebra and geometry to arbitrary noncommutative rings.