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We present an effective method to compute the resolution regularity (vector) of bi-homogeneous ideals. For this purpose, we first introduce the new notion of an x-Pommaret basis and describe an algorithm to compute a linear change of coordinates for a given bi-homogeneous ideal such that the new ideal obtained after performing this change possesses a finite x-Pommaret basis. Then, we show that the x-component of the bi-graded regularity of a bi-homogeneous ideal is equal to the x-degree of its x-Pommaret basis (after performing the mentioned linear change of variables). Finally, we introduce the new notion of an ideal in x-quasi stable position and show that a bi-homogeneous ideal has a finite x-Pommaret basis iff it is in x-quasi stable position.
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