Polynomilla

Polynomilla is a hypothetical algebraic construct introduced in speculative mathematical literature as a generalization of ordinary polynomials. It models finite linear combinations of monomials with possibly non-integer exponents. Formally, over a commutative ring R and in variables x1, ..., xn, a Polynomilla is a finite sum ∑ a_i ∏_{j=1}^n x_j^{r_{ij}}, where each coefficient a_i lies in R and each exponent vector (r_{i1},…,r_{in}) consists of rational numbers (often drawn from a fixed subset such as nonnegative rationals).

Operations on Polynomilla follow the same basic rules as for polynomials. Addition combines like exponent vectors, and multiplication adds exponent vectors. The notion of degree extends to the maximal total exponent or to a chosen weight function; in practice, a Polynomilla is typically restricted to finite support to ensure well-defined multiplication and truncation. Ordinary polynomials are recovered when all exponents are nonnegative integers and the number of variables is one.

Examples include p(x) = 2 + 3 x^{1/2} + x^{3/2} in one variable, and p(x,y) = 5 x^{1/3} y^{2/3} + 7 in two variables. Because the exponents may be fractional, Polynomillas can model objects with fractional growth or tangency orders, offering a way to illustrate phenomena not captured by traditional polynomials. In algebraic geometry and related areas, Polynomilla-like forms are discussed as simplified tools for exploring the effects of non-integer degrees on curve behavior; in educational or fictional contexts they serve as stepping stones toward Puiseux polynomials and other generalized power products.

Origin and usage: The term Polynomilla appears in speculative mathematical writing and is not part of established standard algebra. It is presented here as a concise, neutral description of a generalized polynomial-like object for conceptual illustration.