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New parameters of subsets in polynomial schemes

We define new parameters, a zero interval and a dual zero interval, of subsets in $P$- or $Q$-polynomial schemes. A zero interval of a subset in a $P$-polynomial scheme is a successive interval index for which the inner distribution vanishes, and a dual zero interval of a subset in a $Q$-polynomial scheme is a successive interval index for which the dual inner distribution vanishes. We derive the bounds of the lengths of a zero interval and a dual zero interval using the degree and dual degree respectively, and show that a subset in a $P$-polynomial scheme (resp. a $Q$-polynomial scheme) having a large length of a zero interval (resp. a dual zero interval) induces a completely regular code (resp. a $Q$-polynomial scheme). Moreover, we consider the spherical analogue of a dual zero interval.