{"id":396356,"date":"2019-01-17T09:56:05","date_gmt":"2019-01-17T17:56:05","guid":{"rendered":"https:\/\/www.microsoft.com\/en-us\/research\/?post_type=msr-research-item&#038;p=396356"},"modified":"2019-01-17T09:56:05","modified_gmt":"2019-01-17T17:56:05","slug":"computing-gcds-polynomials-algebraic-number-fields","status":"publish","type":"msr-research-item","link":"https:\/\/www.microsoft.com\/en-us\/research\/publication\/computing-gcds-polynomials-algebraic-number-fields\/","title":{"rendered":"Computing GCDs of Polynomials over Algebraic Number Fields"},"content":{"rendered":"<div class=\"Abstracts\">\n<div id=\"aep-abstract-id3\" class=\"abstract author\">\n<div id=\"aep-abstract-sec-id4\">\n<p>Modular methods for computing the gcd of two univariate polynomials over an algebraic number field require\u00a0<em>a priori<\/em>\u00a0knowledge about the denominators of the rational numbers in the representation of the gcd. A multiplicative bound for these denominators is derived without assuming that the number generating the field is an algebraic integer. Consequently, the gcd algorithm of Langemyr and McCallum [<em>J. Symbolic Computation<\/em><strong>8<\/strong>, 429 &#8211; 448, 1989] can now be applied directly to polynomials that are not necessarily represented in terms of an algebraic integer. Worst-case analyses and experiments with an implementation show that by avoiding a conversion of representation the reduction in computing time can be significant. A further improvement is achieved by using an algorithm for reconstructing a rational number from its modular residue so that the denominator bound need not be explicitly computed. Experiments and analyses suggest that this is a good practical alternative.<\/p>\n<\/div>\n<\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Modular methods for computing the gcd of two univariate polynomials over an algebraic number field require\u00a0a priori\u00a0knowledge about the denominators of the rational numbers in the representation of the gcd. A multiplicative bound for these denominators is derived without assuming that the number generating the field is an algebraic integer. Consequently, the gcd algorithm of [&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-396356","msr-research-item","type-msr-research-item","status-publish","hentry","msr-research-area-computational-sciences-mathematics","msr-locale-en_us"],"msr_publishername":"Academic 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