= Remez algorithm
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The Remez algorithm searches for a <best uniform approximation> by solving alternation equations on a reference set, finding extrema of the error, and exchanging reference points. For real degree-at-most-$n$ <polynomials>, the <equioscillation theorem> characterizes success by $n+2$ ordered alternating maximal errors. The <alternating-reference error bound> supplies a certified lower bound at each step, while the largest absolute error supplies an upper bound. Their gap measures how close the current <polynomial> is to optimal, provided the upper bound is established globally rather than only on a finite grid.
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