{"id":158205,"date":"2005-07-01T00:00:00","date_gmt":"2005-07-01T00:00:00","guid":{"rendered":"https:\/\/www.microsoft.com\/en-us\/research\/msr-research-item\/solvability-of-a-system-of-bivariate-polynomial-equations-over-a-finite-field\/"},"modified":"2018-10-16T20:14:51","modified_gmt":"2018-10-17T03:14:51","slug":"solvability-of-a-system-of-bivariate-polynomial-equations-over-a-finite-field","status":"publish","type":"msr-research-item","link":"https:\/\/www.microsoft.com\/en-us\/research\/publication\/solvability-of-a-system-of-bivariate-polynomial-equations-over-a-finite-field\/","title":{"rendered":"Solvability of a System of Bivariate Polynomial Equations over a Finite Field"},"content":{"rendered":"<div class=\"asset-content\">\n<p>We study the following problem &#8211; given a finite field F and a set of polynomials p1(x), p2(x), &#8230; , pm(x) in n variables x=(x1, &#8230;, xn) over F, determine if the corresponding system of equations p1(x) = p2(x) = &#8230; = pm(x) = 0 has a solution over F or not. We investigate the complexity of this problem when the number of variables is bounded. We show that for any fixed value of n, there is a deterministic algorithm for deciding solvability whose running time is bounded by a polynomial in d, m and log |F|. Previously, only a randomized polynomial-time algorithm was known (Huang and Wong, FOCS 1996). In particular, this result also leads to a deterministic algorithm for recognizing permutation functions, a problem which was previously known to be in ZPP (Ma and Gathen, STOC 1994) but not known to be in P.<\/p>\n<\/div>\n<p><!-- .asset-content --><\/p>\n","protected":false},"excerpt":{"rendered":"<p>We study the following problem &#8211; given a finite field F and a set of polynomials p1(x), p2(x), &#8230; , pm(x) in n variables x=(x1, &#8230;, xn) over F, determine if the corresponding system of equations p1(x) = p2(x) = &#8230; = pm(x) = 0 has a solution over F or not. We investigate the [&hellip;]<\/p>\n","protected":false},"featured_media":0,"template":"","meta":{"msr-url-field":"","msr-podcast-episode":"","msrModifiedDate":"","msrModifiedDateEnabled":false,"ep_exclude_from_search":false,"_classifai_error":"","msr-author-ordering":[{"type":"user_nicename","value":"neeraka","user_id":"33076"}],"msr_publishername":"Springer","msr_publisher_other":"","msr_booktitle":"","msr_chapter":"","msr_edition":"International Colloquium on Automata, Languages and Programming (ICALP)","msr_editors":"","msr_how_published":"","msr_isbn":"3-540-27580-0","msr_issue":"","msr_journal":"","msr_number":"","msr_organization":"","msr_pages_string":"551\u2013562","msr_page_range_start":"551","msr_page_range_end":"562","msr_series":"Lecture Notes in Computer Science","msr_volume":"3580","msr_copyright":"","msr_conference_name":"International Colloquium on Automata, Languages and Programming 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