Concepts inCombinatorics of 4-dimensional resultant polytopes
Resultant
In mathematics, the resultant of two univariate polynomials and is a polynomial function of their coefficients that is zero if and and only if the two polynomials have a common root in an algebraically closed field containing the coefficients. Alternatively the resultant is sometimes defined for two homogeneous polynomials in two variables, in which case it vanishes when the polynomials have a common non-zero solution, or equivalently when they have a common zero on the projective line.
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Polytope
In elementary geometry, a polytope is a geometric object with flat sides, which exists in any general number of dimensions. A polygon is a polytope in two dimensions, a polyhedron in three dimensions, and so on in higher dimensions (such as a polychoron in four dimensions). Some theories further generalize the idea to include such things as unbounded polytopes, and abstract polytopes. When referring to an n-dimensional generalization, the term n-polytope is used.
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Combinatorics
Combinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Aspects of combinatorics include counting the structures of a given kind and size, deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria, finding "largest", "smallest", or "optimal" objects, and studying combinatorial structures arising in an algebraic context, or applying algebraic techniques to combinatorial problems.
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Polyhedral combinatorics
Polyhedral combinatorics is a branch of mathematics, within combinatorics and discrete geometry, that studies the problems of counting and describing the faces of convex polyhedra and higher dimensional convex polytopes. Research in polyhedral combinatorics falls into two distinct areas.
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Newton polygon
In mathematics, the Newton polygon is a tool for understanding the behaviour of polynomials over local fields. In the original case, the local field of interest was the field of formal Laurent series in the indeterminate X, i.e. the field of fractions of the formal power series ring K, over K, where K was the real number or complex number field. This is still of considerable utility with respect to Puiseux expansions.
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Facet (mathematics)
A facet of a simplicial complex is a maximal simplex. In the general theory of polyhedra and polytopes, two conflicting meanings are currently jostling for acceptability: A facet of a geometric polyhedron is traditionally any polygon whose corners are vertices of the polyhedron. By extension to higher dimensions, it is any j-tope whose vertices are shared by some n-tope (n-dimensional polytope where 0 < j < n).
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Face (geometry)
In geometry, a face of a polyhedron is any of the polygons that make up its boundaries. For example, any of the squares that bound a cube is a face of the cube. The suffix -hedron is derived from the Greek word ¿¿¿¿ (hedra) which means "face". Sometimes, in the case of a pyramid, the term face is understood to exclude the base. The (two-dimensional) polygons that bound higher-dimensional polytopes are also commonly called faces.
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Degree of a polynomial
The degree of a polynomial is the highest degree of its terms, when the polynomial is expressed in canonical form (i.e. as a linear combination of monomials). The degree of a term is the sum of the exponents of the variables that appear in it. The word degree is now standard, but in some older books, the word order may be used instead. For example, the polynomial has three terms. (Notice, this polynomial can also be expressed as .
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