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We compute the $F$-pure threshold of non necessarily principal ideals whichsatisfy a geometric generic condition about their Newton polygons. We alsocontribute some evidence in favor of the conjectured equality between the$F$-pure threshold and the log canonical threshold of ideals. These results areobtained by generalizing the theory of splitting polytopes to the case ofideals. As applications of our results we obtain geometric lower bounds for therecently introduced $F$-volume of a collection of ideals.

$F$-pure thresholds and $F$-Volumes of some non principal ideals