Theoretical Computer Science 1I (1980) 321--330 @ North-Holland Publishing Company WER BOUNDS FOR POLYNOMIALS WITH ALGEBRA COEFFICIENTS Joos HEINTZ and Malte SIEVEKING FachbereichMathematik, Johann- Wolfgang Goethe Universitiit, Frankfurt am Main, F.R.G. Communicated by A. Schiinhage Received September 1978 Revised May 1979 Abstract. We give a method, based on algebraic geometry, to show lower bounds for the complexity of polynomi4s with algebraic coefficients. Typical examples are polynomials wit-h coefficients which are roots of unity, such as IZe Zri/iXi and i e2’i‘i/&Xj, zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLK j-l j=l where pi is the fh prime number. We apply the method also to systems of linear equations. It is well known by results of Motzkin [7], Belaga [ 1] and Paterson and Stockmeyer [8] that ‘in general’ the evaluation of a polynomial zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQP F(X) = zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA fdXd + 9 l l + fo E C[X] needs d additions/subtractions, approximately id scalar multiplications, and zyxwvutsrqpon order Jd nonscalar multiplications. On the other hand, F(X) can be computed with this amount of operations. However, it was Strassen [ 121 who gave concrete examples of polynomials with algebraic and rational coefficients, which are hard to compute. This work was continued by Borodin and Cook [2], Schnorr [9] and Schnorr and van der Wiele [lo]. In this paper we translate complexity theory into algebraic geometry. To any polynomial F with deg F = d and coefficients from an algebraically closed field fi WP associate a morphism A m h zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA Ad of affine spaces over A, where m depends on the complexity of F. Let W be the Zariski closure of the image of t,4 in Ad. At the beginning, in Lemma 1, we state a connection between the complexity of F, deg F, and deg W. Then, in Theorem 1, we give a lower bound for deg W in terms of the coefficients of F. The bound is nontrivial if for example A= C and the coefficients of F are algebraic of high degree over 321