{"id":475,"date":"2023-01-23T15:35:05","date_gmt":"2023-01-23T15:35:05","guid":{"rendered":"https:\/\/math-sites.uncg.edu\/number-theory\/?p=475"},"modified":"2025-01-24T19:37:57","modified_gmt":"2025-01-24T19:37:57","slug":"zeros-of-fractional-derivatives-of-polynomials","status":"publish","type":"post","link":"https:\/\/math-sites.uncg.edu\/number-theory\/zeros-of-fractional-derivatives-of-polynomials\/","title":{"rendered":"Zeros of Fractional Derivatives of Polynomials"},"content":{"rendered":"<p>\nWe consider the polynomial p(x)=(x+1)(x-2-i)(x-3-i)(x-1-i) over the complex numbers.  We have deg p\u2032(x) = deg p(x)\u22121 = 4-1 = 3.  The Fundamental Theorem of Algebra asserts that the same reduction occurs for the total number of zeros of the polynomials.  When we consider the fractional derivatives the following questions arise:<\/p>\n<ul>\n<li>What happens to the zeros of fractional derivatives, as the real \u03b1 increases continuously from 0 to n ?<\/li>\n<li>How do the zeros of polynomials vanish ?<\/li>\n<\/ul>\n<p>\nThe video below shows the \u03b1-th Riemann-Liouville fractional derivative of the polynomial (x+1)(x-2-i)(x-3-i)(x-1-i) for -1&lt; \u03b1 &lt; 9.\n<\/p>\n<p>\nThe hue represents the argument with red representing the positive real direction and cyan the negative real direction, as shown on the right. Brightness represents absolute value, with 0 represented by black and with white representing infinity.\n<\/p>\n<p>\nPlots were generated with <a href=\"https:\/\/www.sagemath.org\">SageMath<\/a>.  The video was complied with <a href=\"https:\/\/ffmpeg.org\">ffmpeg<\/a>.\n<\/p>\n<p>\nFor more see, <a href=\"https:\/\/math-sites.uncg.edu\/sites\/pauli\/publications\/Caparatta-Pauli-Saidak_Fractional-derivatives-of-polynomials.pdf\">Zeros of Fractional Derivatives of Polynomials<\/a> by Torre Caparatta, Sebastian Pauli and Filip Saidak\n<\/p>\n<p><iframe loading=\"lazy\" title=\"Riemann-Liouville fractional derivatives of a degree 4 polynomial\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/TJMG4tFvHSA?feature=oembed\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" referrerpolicy=\"strict-origin-when-cross-origin\" allowfullscreen><\/iframe><\/p>\n<p>\nWe show the paths of the roots in the following plot.\n<\/p>\n<p><a href=\"\"https:\/\/math-sites.uncg.edu\/number-theory\/wp-content\/uploads\/sites\/6\/2023\/01\/x1x-2-ix-3-ix-1-i__all_0.png\"><br \/>\n<image width=480px\" src=\"https:\/\/math-sites.uncg.edu\/number-theory\/wp-content\/uploads\/sites\/6\/2023\/01\/x1x-2-ix-3-ix-1-i__all_0.png\"\/><\/a><\/p>\n","protected":false},"excerpt":{"rendered":"<p>We consider the polynomial p(x)=(x+1)(x-2-i)(x-3-i)(x-1-i) over the complex numbers. We have deg p\u2032(x) = deg p(x)\u22121 = 4-1 = 3. The Fundamental&#8230;<\/p>\n","protected":false},"author":12,"featured_media":492,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[8],"tags":[],"class_list":["post-475","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-number-theory"],"acf":[],"_links":{"self":[{"href":"https:\/\/math-sites.uncg.edu\/number-theory\/wp-json\/wp\/v2\/posts\/475","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/math-sites.uncg.edu\/number-theory\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/math-sites.uncg.edu\/number-theory\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/math-sites.uncg.edu\/number-theory\/wp-json\/wp\/v2\/users\/12"}],"replies":[{"embeddable":true,"href":"https:\/\/math-sites.uncg.edu\/number-theory\/wp-json\/wp\/v2\/comments?post=475"}],"version-history":[{"count":21,"href":"https:\/\/math-sites.uncg.edu\/number-theory\/wp-json\/wp\/v2\/posts\/475\/revisions"}],"predecessor-version":[{"id":502,"href":"https:\/\/math-sites.uncg.edu\/number-theory\/wp-json\/wp\/v2\/posts\/475\/revisions\/502"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/math-sites.uncg.edu\/number-theory\/wp-json\/wp\/v2\/media\/492"}],"wp:attachment":[{"href":"https:\/\/math-sites.uncg.edu\/number-theory\/wp-json\/wp\/v2\/media?parent=475"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/math-sites.uncg.edu\/number-theory\/wp-json\/wp\/v2\/categories?post=475"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/math-sites.uncg.edu\/number-theory\/wp-json\/wp\/v2\/tags?post=475"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}