Lectures on the complexity of bilinear problems
Lectures on the complexity of bilinear problems
Inverting polynomials and formal power series
SIAM Journal on Computing
Bivariate Polynomial Multiplication Patterns
AAECC-11 Proceedings of the 11th International Symposium on Applied Algebra, Algebraic Algorithms and Error-Correcting Codes
Bivariate Polynomial Multiplication
FOCS '98 Proceedings of the 39th Annual Symposium on Foundations of Computer Science
Multiplicative Complexity of Taylor Shifts and a New Twist of the Substitution Method
FOCS '98 Proceedings of the 39th Annual Symposium on Foundations of Computer Science
Algebraic Complexity Theory
Beyond the Alder-Strassen bound
Theoretical Computer Science - Automata, languages and programming
Multivariate power series multiplication
Proceedings of the 2005 international symposium on Symbolic and algebraic computation
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Inspired by Schönhage's discussion in the Proc. 11th Applied Algebra and Error Correcting Codes Conference (AAECC), Lecture Notes in Comput. Sci., Springer, Berlin, Vol. 948, 1995 pp. 70, we study the multiplicative complexity of the multiplication, squaring, inversion, and division of bivariate power series modulo the "triangular" and "quadratic" ideals (Xd+1, XdY,Xd-1 Y2,..., Yd+1) and (Xd+1, Yd+1), respectively. For multiplication, we obtain the lower bounds 5/4 d2 - O(d) and 2 1/3 d2 O(d) for the triangular and quadratic case, respectively, opposed to the upper bounds 3/2 d2 + O(d) and 3d2 + O(d). For squaring, we prove the lower bounds 7/8 d2-O(d) and 1 3/5 d2- O(d). As upper bounds, we have d2+O(d) and 2 ½ d2+O(d) for the triangular and quadratic case, respectively. Concerning inversion, the obtained lower bounds coincide with those of squaring. As upper bounds, we show 3 5/6 d2 + O(d) and 8 1/3 d2 + O(d), respectively. The lower bounds for division are those of multiplication. The upper bounds follow from combining the bounds for inversion and multiplication. All of the above lower bounds hold over arbitrary fields (in the case of multiplication and division) and over fields of characteristic distinct from two (in the case of squaring and inversion), respectively. All upper bounds are valid for fields that "support FFTs", that is, fields that have characteristic zero and contain all roots of unity.