Query to find the Intersection coordinates of multiple Polyon

The ST_Intersect function works with only two geometries at a time. It is a binary relation only.

As katahdin wrote, ST_Intersection works only with two geometries. I don't know a variant of it what works on arrays. If you have your geometries in one table, you can write a function that will do the job, like this:

create or replace function ST_IntersectionArray(geoms geometry[]) returns geometry as $$
declare
   i integer;
   tmpGeom geometry;
begin
    tmpGeom := geoms[1];
    FOR i IN 1..array_length(geoms,1) LOOP
      tmpGeom:= ST_Intersection(tmpGeom,geoms[i]);
    END LOOP;
    return tmpGeom;
end;
$$
LANGUAGE plpgsql;

Example call:

SELECT ST_AsText(ST_IntersectionArray(ARRAY(select geom from test_geom)));

If your polygons are in two separate tables, say poly1, and poly2, then

SELECT As_Text(St_Intersection(poly1.geom, poly2.geom))
FROM poly1, poly2;

If all the polygons are in one table, polys, then, assuming you have a primary key 'id':

SELECT As_Text(ST_Intersection(p1.geom, p2.geom))
FROM polys AS p1, polys AS p2
WHERE p1.id <> p2.id