subtractionfree

Subtractionfree is an adjective used in mathematics to describe expressions, formulas, or computational procedures that do not involve subtraction. In subtractionfree constructions, all operations are additions, multiplications, and sometimes divisions, with the aim that the result remains nonnegative whenever the inputs are nonnegative.

The concept is tied to positivity in algebra and number theory. Subtractionfree forms are valued because they preserve positivity under evaluation and often admit constructive proofs of positivity. They appear in areas such as combinatorics, representation theory, and algebraic geometry, where subtraction-free identities can provide stable, overflow-avoiding ways to compute with positive quantities and to demonstrate that certain expressions remain nonnegative.

In more specialized settings, subtraction-free expressions arise in the theory of cluster algebras and related combinatorial frameworks. Many cluster variables can be written as subtraction-free Laurent expressions in any chosen cluster, a feature connected to the Laurent phenomenon and to positivity results in these theories. Subtraction-free representations also surface in the study of totally positive matrices and in the design of algorithms that operate on inherently positive data without introducing negative intermediate values.

Examples of subtractionfree expressions include a+b, ab, (a+b)c, and a/(b+c) (when division is allowed). Expressions that necessarily involve subtraction, such as a−b, are not subtractionfree unless rewritten through identities that avoid subtraction. Limitations exist, as not all quantities admit subtraction-free forms, and some transformations can increase complexity or affect numerical behavior.