Coordinate-wise valuation is a function that assigns a value to each coordinate of a point in a given space, typically reflecting some kind of measure of 'size' or 'importance' of that coordinate. This concept is central to the tropicalization process, where we translate classical algebraic varieties into a tropical setting by focusing on the minimum values among coordinates. It helps in understanding the geometry of these varieties by enabling comparisons and constructions based on the valuations of their coordinates.
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