Interval rational = algebraic

Abstract

Rational functions can be defined as compositions of arithmetic operations (+, -,.,:). What class of functions will be get if we add to this list the "interval" operation (that transforms a function f of n variables and given intervals X1, ...., Xn into the bounds for the range f(X1, ..., Xn))? In this paper, we prove that adding this "interval" operation to rational functions leads exactly to the set of all (locally) algebraic functions.In other words, algebraic functions can be described as compositions of arithmetic operations and the "interval" operation.This result provides an additional explanation of why naive interval computations sometimes overshoot:• the desired dependency is (locally) a genera algebraic function;• naive interval methods results in a (locally) rational function;• not all algebraic functions are rational.