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I am looking for the PostGIS ST_function that will give me the same results as running the ArcGIS Analyis-> Overlays-> Intersect tool on two polygons.

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What your looking for is ST_Intersection.

ST_Intersection — (T) Returns a geometry that represents the shared portion of geomA and geomB. The geography implementation does a transform to geometry to do the intersection and then transform back to WGS84.

geometry ST_Intersection( geometry geomA , geometry geomB );

You can see a query that implements it in this question (down at the bottom).

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