Homogeneous polynomial

A polynomial is homogeneous of degree

n

when scaling all variables by

λ

scales its value by

λ^{n}

.

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- Monomial containment in an ideal of coordinate powers Homogeneous polynomial

Monomial containment in an ideal of coordinate powers ( $(X_{0}^{d_{0}}, \dots, X_{n}^{d_{n}}) \supseteq k [X_{0}, \dots, X_{n}]_{\sum d_{i}}$ )

d_{i} \geq 0

, every monomial of total degree

\sum_{i} d_{i}

is divisible by some

X_{i}^{d_{i}}

. If any

d_{i} = 0

the ideal is the whole ring. Otherwise a monomial avoiding all these divisibilities would have total degree at most

\sum_{i} (d_{i} - 1)

, a contradiction. The sharper least degree containing every monomial is

\sum_{i} (d_{i} - 1) + 1

when all

d_{i} > 0

.

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Homogeneous polynomial by

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A **homogeneous polynomial** is a polynomial whose terms all have the same total degree. In more formal terms, a polynomial \( P(x_1, x_2, \ldots, x_n) \) is called homogeneous of degree \( d \) if every term in the polynomial is of degree \( d \).

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