Shape (algebra)

The form is a term from the mathematical branch of algebra . Shapes are special homogeneous polynomials .

definition

A shape is a polynomial in several variables where every summand has the same degree .

Examples and special cases

• The polynomial of degree three in three variables is a shape. ${\displaystyle X^{3}+Y^{3}+Z^{3}+XY^{2}+XYZ}$
• A polynomial in a variable of degree one is called a linear form , since these polynomials can be viewed as linear mappings .
• A polynomial in two variables of degree one is called a bilinear form . However, a sesquilinear form that is very similar to the bilinear form cannot be understood as a form within the meaning of the article.
• A polynomial in variables of degree one is called a multilinear form . The elements of outer algebra are a special class of multilinear forms . These are generalized to differential forms in differential geometry . ${\displaystyle n}$
• A polynomial in variables of degree two is called a quadratic form . A special class of square shapes are the binary square shapes . These are the polynomials of degree two in two variables. ${\displaystyle n}$

literature

• Shape . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).