Lower bounds for polynomial evaluation and interpolation problems

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We show that there is a set of pointsp 1,p 2,...,p n such that any arithmetic circuit of depthd for polynomial evaluation (or interpolation) at these points has size $$\Omega \left( {\frac{{n\log n}}{{\log (2 + d/\log n}}} \right).$$ Moreover, for circuits of sub-logarithmic depthd, we obtain a lower bound of Ω(dn 1+1/d ) on its size.