Zerocoefficient

Zerocoefficient is a term used in mathematics and related fields to denote a coefficient that equals zero in a linear combination, polynomial, or series. When a coefficient is zero, the corresponding term has no contribution to the evaluated expression, and the term may be omitted from a compact representation. The concept emphasizes sparsity of a representation rather than a value change.

In polynomials or power series, if a_k denotes the coefficient of x^k in a polynomial p(x) = sum a_k x^k, then a_k = 0 means the x^k term is absent from the expanded form, even though the expression may include many other nonzero coefficients. In linear algebra, a coefficient vector with zeros indicates components that do not affect the resulting vector under the given basis; zero entries contribute to sparsity.

In practice, zero coefficients arise through simplification, factoring, or as a result of modeling choices. In signal processing, a zero coefficient in a Fourier or discrete cosine transform corresponds to the absence of that frequency component. In statistics and machine learning, regularization methods such as L1 regularization encourage models where many coefficients are exactly zero, yielding sparse, interpretable representations.

The term zerocoefficient is typically used informally; precise language usually distinguishes zero coefficients from nonzero coefficients or from roots. The concept underlies many ideas of sparsity and efficient representation across mathematics, computer science, and engineering.