{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T18:39:36Z","timestamp":1787337576284,"version":"build-2736575974"},"reference-count":22,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Comput."],"published-print":{"date-parts":[[2005,1]]},"abstract":"<jats:p>Let R(A) denote the rank (also called bilinear complexity) of a finite dimensional associative algebra A. A fundamental lower bound for R(A) is the so-called Alder--Strassen bound R(A) \\ge 2 \\dim A - t, where t is the number of maximal twosided ideals of A. An algebra is called an algebra of minimal rank if the Alder--Strassen bound is tight, i.e., $R(A) = 2 \\dim A - t.<\/jats:p>\n                  <jats:p>As the main contribution of this work, we characterize all algebras of minimal rank over arbitrary fields. This finally solves an open problem in algebraic complexity theory; see, for instance, [V. Strassen, Handbook of Theoretical Computer Science, J. van Leeuwen, ed., Elsevier Science, New York, 1990, Vol.\\ A, pp. 634--672, section 12, Problem 4] or [P. B\u00fcrgisser, M. Clausen, and M. A. Shokrollahi, Algebraic Complexity Theory, Springer, New York, 1997, Problem 17.5].<\/jats:p>","DOI":"10.1137\/s0097539703438277","type":"journal-article","created":{"date-parts":[[2004,12,9]],"date-time":"2004-12-09T21:00:32Z","timestamp":1102626032000},"page":"277-298","source":"Crossref","is-referenced-by-count":7,"title":["A Complete Characterization of the Algebras of Minimal Bilinear Complexity"],"prefix":"10.1137","volume":"34","author":[{"given":"Markus","family":"Bl\u00e4ser","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2006,7,27]]},"reference":[{"key":"R1","doi-asserted-by":"publisher","DOI":"10.1016\/0304-3975(81)90070-0"},{"key":"R2","doi-asserted-by":"publisher","DOI":"10.1016\/0304-3975(91)90017-V"},{"key":"R3","doi-asserted-by":"publisher","DOI":"10.1007\/s000370050028"},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.1007\/PL00001605"},{"key":"R5","doi-asserted-by":"crossref","unstructured":"M. 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R\u00f3nyai,\n                      Polynomial time solutions of some problems in abstract algebra\n                      , in Proceedings of the 17th Annual ACM Symposium on Theory of Computing (STOC), Providence, RI, 1985, ACM, New York, 1985, pp. 153\u2013162.","DOI":"10.1145\/22145.22162"},{"key":"R14","doi-asserted-by":"publisher","DOI":"10.1137\/0212007"},{"key":"R15","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0020719"},{"key":"R16","doi-asserted-by":"publisher","DOI":"10.1016\/0024-3795(83)90165-9"},{"key":"R17","doi-asserted-by":"publisher","DOI":"10.1137\/0214029"},{"key":"R18","doi-asserted-by":"crossref","unstructured":"Joos Heintz, Jacques Morgenstern, On associative algebras of minimal rank, Lecture Notes in Comput. 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