Coefficientilla

Coefficientilla is a scalar associated with a polynomial that serves as a compact descriptor of the typical size of its coefficients after normalizing the polynomial by its leading coefficient. It is defined to be invariant under nonzero scalar multiplication, making it convenient for comparing families of polynomials that differ by an overall factor.

For a nonzero polynomial P(x) = a_d x^d + ... + a_0 with a_d ≠ 0, the coefficientilla C(P) is defined as C(P) = (1/(d+1)) sum_{k=0}^d |a_k / a_d|. Equivalently, if you scale P by 1/a_d to make the leading coefficient 1, C(P) is the average of the absolute values of the resulting coefficients.

Example 1: P(x) = x^3 + 2x^2 - 5x + 7. Then a_d = 1, and C(P) = (|1| + |2| + |−5| + |7|)/4 = 15/4 = 3.75.

Example 2: Q(x) = 3x^2 + x − 4. Then a_d = 3, coefficients normalized: (−4/3, 1/3, 1). C(Q) = (|−4/3| + |1/3| + |1|)/3 = (4/3 + 1/3 + 1)/3 = 8/9 ≈ 0.888.

Extensions and variants include using other norms, such as an L2-based coefficientilla C2(P) = sqrt(sum (a_k / a_d)^2) / sqrt(d+1), which provides a root-mean-square measure of the normalized coefficients.

Uses and interpretation: coefficientilla offers a scale-invariant way to compare coefficient magnitudes across polynomial families and can help in discussions of sparsity, normalization, and the effect of algebraic operations on coefficient sizes. It is a fictional construct used for illustrative purposes and is not a standard term in established literature. See also coefficients, polynomial norms, and sparsity measures.

---