Order of a Polynomial

Summary: Euler defines the order of a polynomial as the greatest degree of any term appearing in it. In multivariate settings this is the classification tied to algebraic curves and surfaces.

Sources: chapter5

Last updated: 2026-04-24

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Definition

The order of a polynomial is the greatest total degree of any single term that appears in it (source: chapter5, §94).

Thus $z^2+y^2+z^2+ay-a^2$ is of order 2, while $y^4+y{z}^3-a{y}^{2}x+abyz-a^2{y}^2+b^4$ is of order 4 (source: chapter5, §94).

Order versus degree

For a homogeneous polynomial, all terms have the same degree, so order and degree coincide.

For a heterogeneous polynomial, lower-degree terms may appear alongside the highest-degree ones; the order records only the greatest degree present (source: chapter5, §94).

Why Euler uses it

Euler says this is the classification relevant to the study of algebraic curves: second-order curves are conics, third-order curves are cubics, and so on (source: chapter5, §94).

Related pages

• reducible-polynomial
• homogeneous-function
• heterogeneous-function
• chapter-5-on-functions-of-two-or-more-variables