{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,18]],"date-time":"2026-09-18T19:15:44Z","timestamp":1789758944685,"version":"4.0.1"},"reference-count":32,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"1","funder":[{"DOI":"10.13039\/501100001852","name":"CEFIPRA","doi-asserted-by":"crossref","award":["4702-1"],"award-info":[{"award-number":["4702-1"]}],"id":[{"id":"10.13039\/501100001852","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Comput."],"published-print":{"date-parts":[[2017,1]]},"abstract":"<jats:p>We show here a $2^{\\Omega(\\sqrt{d} \\cdot \\log N)}$ size lower bound for homogeneous depth four arithmetic formulas over fields of characteristic zero. That is, we give an explicit family of polynomials of degree $d$ on $N$ variables (with $N = d^3$ in our case) with 0, 1-coefficients such that for any representation of a polynomial $f$ in this family of the form $ f = \\sum_{i} \\prod_{j} Q_{ij}, $ where the $Q_{ij}$'s are homogeneous polynomials (recall that a polynomial is said to be homogeneous if all its monomials have the same degree), it must hold that $ \\sum_{i, j} (\\text{number of monomials of~} Q_{ij}) \\geq 2^{\\Omega (\\sqrt{d} \\cdot \\log N)}. $ The abovementioned family, which we refer to as the Nisan--Wigderson design-based family of polynomials, is in the complexity class $\\mathsf{VNP}$. Our work builds on recent lower bound results and yields an improved quantitative bound as compared to the quasi-polynomial lower bound of [N. Kayal et al., in Symposium on Theory of Computing, ACM, New York, 2014, pp. 119--127] and the $N^{\\Omega(\\log \\log N)}$ lower bound in the independent work of [M. Kumar and S. Saraf, in Automata, Languages, and Programming, Part I, Springer, Berlin, 2014, pp. 751--762].<\/jats:p>","DOI":"10.1137\/151002423","type":"journal-article","created":{"date-parts":[[2017,2,28]],"date-time":"2017-02-28T11:29:24Z","timestamp":1488281364000},"page":"307-335","source":"Crossref","is-referenced-by-count":13,"title":["An Exponential Lower Bound for Homogeneous Depth Four Arithmetic Formulas"],"prefix":"10.1137","volume":"46","author":[{"given":"Neeraj","family":"Kayal","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Nutan","family":"Limaye","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Chandan","family":"Saha","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Srikanth","family":"Srinivasan","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2017,2,28]]},"reference":[{"key":"atypb1","doi-asserted-by":"publisher","DOI":"10.1017\/S0963548307008917"},{"key":"atypb2","doi-asserted-by":"crossref","unstructured":"M. Agrawal, C. Saha, R. Saptharishi, and N. Saxena,\n                      Jacobian hits circuits: Hitting-sets, lower bounds for depth-d occur-k formulas & depth-$3$ transcendence degree-k circuits\n                      , in Proceedings of the 44th Symposium on Theory of Computing Conference, STOC 2012, New York, ACM, New York, 2012, pp. 599-614.","DOI":"10.1145\/2213977.2214033"},{"key":"atypb3","first-page":"67","author":"Agrawal M.","year":"2008","journal-title":"NJ"},{"key":"atypb4","doi-asserted-by":"crossref","unstructured":"B. Barak, Z. Dvir, A. Yehudayoff, and A. Wigderson,\n                      Rank bounds for design matrices with applications to combinatorial geometry and locally correctable codes\n                      , in Proceedings of the 43rd ACM Symposium on Theory of Computing, STOC 2011, San Jose, CA, ACM, New York, 2011, pp. 519-528.","DOI":"10.1145\/1993636.1993705"},{"key":"atypb5","first-page":"491","author":"Cai J.-y.","year":"2008","journal-title":"New York"},{"key":"atypb6","doi-asserted-by":"publisher","DOI":"10.1561\/0400000043"},{"key":"atypb7","doi-asserted-by":"crossref","unstructured":"Z. Dvir, S. Saraf, and A. Wigderson,\n                      Breaking the quadratic barrier for 3-lcc's over the reals\n                      , in Symposium on Theory of Computing, STOC 2014, New York, ACM, New York, 2014, pp. 784-793.","DOI":"10.1145\/2591796.2591818"},{"key":"atypb8","doi-asserted-by":"crossref","unstructured":"H. Fournier, N. Limaye, G. Malod, and S. Srinivasan,\n                      Lower bounds for depth 4 formulas computing iterated matrix multiplication\n                      , in Symposium on Theory of Computing, STOC 2014, New York, ACM, New York, 2014, pp. 128-135.","DOI":"10.1145\/2591796.2591824"},{"key":"atypb9","first-page":"577","author":"Grigoriev D.","year":"1998","journal-title":"New York"},{"key":"atypb10","doi-asserted-by":"crossref","unstructured":"A. Gupta, P. Kamath, N. Kayal, and R. Saptharishi,\n                      Approaching the chasm at depth four\n                      , in Proceedings of the 28th Conference on Computational Complexity, CCC 2013, Palo Alto, CA, IEEE, Piscataway, NJ, 2013, pp. 65-73.","DOI":"10.1109\/CCC.2013.16"},{"key":"atypb11","first-page":"578","author":"Gupta A.","year":"2013","journal-title":"NJ"},{"key":"atypb12","doi-asserted-by":"crossref","unstructured":"D. Grigoriev and A. A. Razborov,\n                      Exponential complexity lower bounds for depth 3 arithmetic circuits in algebras of functions over finite fields\n                      , in 39th Annual Symposium on Foundations of Computer Science, FOCS '98, Palo Alto, CA, IEEE Computer Society, Los Alamitos, CA, 1998, pp. 269-278.","DOI":"10.1109\/SFCS.1998.743456"},{"key":"atypb13","first-page":"81","author":"Kayal N.","year":"2012","journal-title":"Electronic Colloquium on Computational Complexity"},{"key":"atypb14","doi-asserted-by":"crossref","unstructured":"N. Kayal, N. Limaye, C. Saha, and S. Srinivasan,\n                      Super-polynomial lower bounds for depth-4 homogeneous arithmetic formulas\n                      , in Symposium on Theory of Computing, STOC 2014, New York, ACM, New York, 2014, pp. 119-127.","DOI":"10.1145\/2591796.2591823"},{"key":"atypb15","doi-asserted-by":"publisher","DOI":"10.1016\/j.tcs.2012.03.041"},{"key":"atypb16","first-page":"68","author":"Kumar M.","year":"2013","journal-title":"Electronic Colloquium on Computational Complexity"},{"key":"atypb17","doi-asserted-by":"crossref","unstructured":"M. Kumar and S. Saraf,\n                      The limits of depth reduction for arithmetic formulas: It's all about the top fan-in\n                      , in Symposium on Theory of Computing, STOC 2014, New York, ACM, New York, 2014, pp. 136-145.","DOI":"10.1145\/2591796.2591827"},{"key":"atypb18","first-page":"364","author":"Kumar M.","year":"2014","journal-title":"NJ"},{"key":"atypb19","first-page":"751","author":"Kumar M.","year":"2014","journal-title":"Berlin"},{"key":"atypb20","doi-asserted-by":"crossref","unstructured":"N. Kayal, C. Saha, and R. 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