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We show how to adapt the Hardy--Littlewood circle method to countmonochromatic solutions to diagonal Diophantine equations. This delivers alower bound which is optimal up to absolute constants. The method isillustrated on equations obtained by setting a diagonal quadratic form equal toa linear form. As a consequence, we determine an algebraic criterion for whensuch equations are partition regular. Our methods involve discrete harmonicanalysis and require a number of `mixed' restriction estimates, which may be ofindependent interest.

Counting monochromatic solutions to diagonal Diophantine equations