{"id":396374,"date":"2019-01-17T09:56:00","date_gmt":"2019-01-17T17:56:00","guid":{"rendered":"https:\/\/www.microsoft.com\/en-us\/research\/?post_type=msr-research-item&#038;p=396374"},"modified":"2019-01-17T09:56:00","modified_gmt":"2019-01-17T17:56:00","slug":"black-box-polynomial-resultants","status":"publish","type":"msr-research-item","link":"https:\/\/www.microsoft.com\/en-us\/research\/publication\/black-box-polynomial-resultants\/","title":{"rendered":"Black-Box Polynomial Resultants"},"content":{"rendered":"<p>A black-box polynomial is a multivariate polynomial that is represented by a program that evaluates the polynomial at an arbitrary point supplied as input. The paper describes an algorithm for constructing a black box for the resultant of two black-box polynomials. The only computationally nontrivial step in the construction is that which determines the degrees of the input black boxes in the variable being eliminated; if those degrees are known, then the black-box resultant can be constructed in a bounded amount of time. Let\u00a0<em>N<\/em>\u00a0be an upper bound for the degrees of the input polynomials. The black-box resultant can be evaluated at an arbitrary point with O(<em>N<\/em>) calls to the input black boxes and O(<em>N<\/em><sup>2<\/sup>) arithmetic operations.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>A black-box polynomial is a multivariate polynomial that is represented by a program that evaluates the polynomial at an arbitrary point supplied as input. The paper describes an algorithm for constructing a black box for the resultant of two black-box polynomials. The only computationally nontrivial step in the construction is that which determines the degrees [&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":"","footnotes":""},"msr-content-type":[3],"msr-research-highlight":[],"research-area":[13546],"msr-publication-type":[193715],"msr-product-type":[],"msr-focus-area":[],"msr-platform":[],"msr-download-source":[],"msr-locale":[268875],"msr-post-option":[],"msr-field-of-study":[],"msr-conference":[],"msr-journal":[],"msr-impact-theme":[],"msr-pillar":[],"class_list":["post-396374","msr-research-item","type-msr-research-item","status-publish","hentry","msr-research-area-computational-sciences-mathematics","msr-locale-en_us"],"msr_publishername":"Elsevier 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