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Let $A$ be an additive basis of order $h$ and $X$ be a finite nonempty subsetof $A$ such that the set $A \setminus X$ is still a basis. In this article, wegive several upper bounds for the order of $A \setminus X$ in function of theorder $h$ of $A$ and some parameters related to $X$ and $A$. If the parameterin question is the cardinality of $X$, Nathanson and Nash already obtained someof such upper bounds, which can be seen as polynomials in $h$ with degree $(|X|+ 1)$. Here, by taking instead of the cardinality of $X$ the parameter definedby $d := \frac{\diam(X)}{\gcd\{x - y | x, y \in X\}}$, we show that the orderof $A \setminus X$ is bounded above by $(\frac{h (h + 3)}{2} + d \frac{h (h -1) (h + 4)}{6})$. As a consequence, we deduce that if $X$ is an arithmeticprogression of length $\geq 3$, then the upper bounds of Nathanson and Nash areconsiderably improved. Further, by considering more complex parameters relatedto both $X$ and $A$, we get upper bounds which are polynomials in $h$ withdegree only 2.

Upper bounds for the order of an additive basis obtained by removing a finite subset of a given basis