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I'm working with QGIS and have a big DEM raster layer with height information and a vector layer with many polygons, representing 1 x 1 degree tiles. Now I want to find for every polygon the highest value of the DEM pixels in it.

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Try SAGA module Shapes - Grid -> Grid statistics for polygons - directly or via QGIS:

Grid statistics for polygons

One of statistics is Maximum - so the result will meet your expectations.

Resulting table will be your input table with new columns named after statistics used in module (as far as I remeber SAGA doesn't obey the 8 character fieldnames limitation for shp - but you can edit it later).

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Is SAGA a conventional plugin, because I don't find it in the plugin list. – Rory Aug 7 '14 at 8:15
Yes, since QGIS 2.0 (I guess) SAGA modules are implemented in QGIS. Check at the menu bar Geoprocessing then open Tools there you'll find SAGA modules. – jzol Aug 7 '14 at 8:20
Thank you very much, it messed the job a little bit up (it created more values than needed with confusing notations), but it has done the important part. :) – Rory Aug 7 '14 at 8:40
In what way did it mess the job? Did it create more columns when you'd chosen only Maximum statistics? As I warned You before: SAGA name fields quite badly (long names like GRIDNAME [statistics name]), then it's shortened to fullify shp standard (8 char fieldname)). Was that the problem? – jzol Aug 21 '14 at 11:42

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