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A Szemeredi-Trotter type theorem, sum-product estimates in finite quasifields, and related results

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Finite field

In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtraction and division are defined and satisfy certain basic rules. The most common examples of finite fields are given by the integers mod p when p is a prime number. The order of a finite field is its number of elements, which is either a prime number or a prime power.

Quasifield

In mathematics, a quasifield is an algebraic structure where and are binary operations on , much like a division ring, but with some weaker conditions. All division rings, and thus all fields, are quasifields. A quasifield is a structure, where and are binary operations on , satisfying these axioms: is a group is a loop, where (left distributivity) has exactly one solution for , Strictly speaking, this is the definition of a left quasifield. A right quasifield is similarly defined, but satisfies right distributivity instead.

Finite ring

In mathematics, more specifically abstract algebra, a finite ring is a ring that has a finite number of elements. Every finite field is an example of a finite ring, and the additive part of every finite ring is an example of an abelian finite group, but the concept of finite rings in their own right has a more recent history. Although rings have more structure than groups, the theory of finite rings is simpler than that of finite groups.

Semifield

In mathematics, a semifield is an algebraic structure with two binary operations, addition and multiplication, which is similar to a field, but with some axioms relaxed. The term semifield has two conflicting meanings, both of which include fields as a special case. In projective geometry and finite geometry (MSC 51A, 51E, 12K10), a semifield is a nonassociative division ring with multiplicative identity element. More precisely, it is a nonassociative ring whose nonzero elements form a loop under multiplication.

Near-field (mathematics)

In mathematics, a near-field is an algebraic structure similar to a division ring, except that it has only one of the two distributive laws. Alternatively, a near-field is a near-ring in which there is a multiplicative identity and every non-zero element has a multiplicative inverse. A near-field is a set together with two binary operations, (addition) and (multiplication), satisfying the following axioms: A1: is an abelian group. A2: = for all elements , , of (The associative law for multiplication).

Planar ternary ring

In mathematics, an algebraic structure consisting of a non-empty set and a ternary mapping may be called a ternary system. A planar ternary ring (PTR) or ternary field is special type of ternary system used by Marshall Hall to construct projective planes by means of coordinates. A planar ternary ring is not a ring in the traditional sense, but any field gives a planar ternary ring where the operation is defined by . Thus, we can think of a planar ternary ring as a generalization of a field where the ternary operation takes the place of both addition and multiplication.

Field (mathematics)

In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational and real numbers do. A field is thus a fundamental algebraic structure which is widely used in algebra, number theory, and many other areas of mathematics. The best known fields are the field of rational numbers, the field of real numbers and the field of complex numbers.

Finite group

In abstract algebra, a finite group is a group whose underlying set is finite. Finite groups often arise when considering symmetry of mathematical or physical objects, when those objects admit just a finite number of structure-preserving transformations. Important examples of finite groups include cyclic groups and permutation groups. The study of finite groups has been an integral part of group theory since it arose in the 19th century.

Field extension

In mathematics, particularly in algebra, a field extension is a pair of fields such that the operations of K are those of L restricted to K. In this case, L is an extension field of K and K is a subfield of L. For example, under the usual notions of addition and multiplication, the complex numbers are an extension field of the real numbers; the real numbers are a subfield of the complex numbers. Field extensions are fundamental in algebraic number theory, and in the study of polynomial roots through Galois theory, and are widely used in algebraic geometry.

Residual sum of squares

In statistics, the residual sum of squares (RSS), also known as the sum of squared residuals (SSR) or the sum of squared estimate of errors (SSE), is the sum of the squares of residuals (deviations predicted from actual empirical values of data). It is a measure of the discrepancy between the data and an estimation model, such as a linear regression. A small RSS indicates a tight fit of the model to the data. It is used as an optimality criterion in parameter selection and model selection.