Multihomogeneous

Multihomogeneous is a term used in mathematics, particularly in algebraic geometry, to describe a certain type of polynomial. A polynomial is considered multihomogeneous if it is homogeneous in each of its variables separately. This means that if you scale each variable by a factor $t_i$ for the $i$-th variable, the polynomial scales by the product of these factors raised to a specific power.

More formally, let $P(x_1, \dots, x_n)$ be a polynomial in $n$ variables. If for each variable $x_i$, there exists a non-negative integer $d_i$ such that $P(x_1, \dots, t_i x_i, \dots, x_n) = t_i^{d_i} P(x_1, \dots, x_n)$ for all scalars $t_i$, then $P$ is multihomogeneous with degrees $(d_1, \dots, d_n)$. A common instance of this is when all $d_i$ are equal, in which case the polynomial is simply homogeneous.

The concept of multihomogeneity is important in the study of algebraic varieties, particularly in defining certain types of geometric objects and their properties. For example, multihomogeneous polynomials appear naturally when studying projective spaces and their embeddings. They are also fundamental in the definition and analysis of Segre embeddings and other constructions in algebraic geometry that involve products of projective spaces. Understanding the multihomogeneous nature of polynomials allows mathematicians to exploit specific symmetries and structures within these geometric contexts.