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In this work we present some conditions of equivalence for the existence of a monomial basis in spaces of homogeneous polynomials on Banach spaces. basis of c 0 is a polynomial-shrinking basis. If E = l 1 , the canonical basis is not a polynomialshrinking basis, since it is not a shrinking basis. (b) Let E = d * (w, 1) be the predual of Lorentz's sequences space. In , Payá and Sevilla have proved that P w ( m d * (w, 1)) = P ( m d * (w, 1) ) if, and only if, w / ∈ l m , where l m is thedoi:10.1016/j.jmaa.2006.09.058 fatcat:j53bp5byxjenvffhfgcrybcwju