On the number of zeros of  harmonic polynomials arranged in alternating  patterns

On the number of zeros of harmonic polynomials arranged in alternating patterns | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article On the number of zeros of harmonic polynomials arranged in alternating patterns Linkui Gao, Gang Liu This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-8698044/v1 This work is licensed under a CC BY 4.0 License Status: Under Review Version 1 posted 6 You are reading this latest preprint version Abstract In this paper, we investigate the number of zeros of harmonic polynomials of the form$$f(z)=a_{1}z^{n_{1}}+a_{2}\overline{z}^{n_{2}}+a_{3}z^{n_{3}}+\cdots+\left(\frac{1+(-1)^{k+1}}{2}a_{k}z^{n_{k}}+\frac{1+(-1)^{k}}{2}a_{k}\overline{z}^{n_{k}}\right)+a_{k+1},$$where $a_{j} (j=1,2\cdots,k+1)$ are non-zero constants and $n_{j} (j=1,2\cdots,k)$ are positive integers with $n_{1}>n_{2}>\cdots>n_{k}\geq1$.Assume that $f$ is regular. Then we show that $f$ has at least $n_1+2(n_2+n_3+\cdots+n_k)$ zeros when each of its corresponding real polynomials has two distinct positive zeros.Furthermore, there are some such harmonic polynomials which have at least $n_1+n_2(n_2+1)$ zeros under particular conditions.In particular, if we choose $n_1=n$ and $n_2=n-1$, then it provides an array of new examples for the sharp bound $n^2$ of zeros of harmonic polynomials $p+\overline{q}$,where $p$ and $q$ are analytic polynomials with degree $n$ and $n-1$ respectively.The sharpness in case $n=3$ is continued to studied, and thus a sufficient condition on two parameters $a$ and $b$ is obtained by Cardano's Formulafor which the harmonic polynomial $f(z)=z^3+a\overline{z}^2+bz+1$ has exactly 9 zeros. Harmonic polynomials Argument Principle for Harmonic Functions Descartes’ Rule of Signs Cardano’s Formula Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Under Review Version 1 posted Reviewers agreed at journal 23 Feb, 2026 Reviewers agreed at journal 09 Feb, 2026 Reviewers invited by journal 04 Feb, 2026 Editor assigned by journal 29 Jan, 2026 Submission checks completed at journal 29 Jan, 2026 First submitted to journal 26 Jan, 2026 You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-8698044","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":585723642,"identity":"49fdd6ab-5ab5-47da-9af3-418717554b73","order_by":0,"name":"Linkui Gao","email":"","orcid":"","institution":"Jiangxi Science and Technology Normal University","correspondingAuthor":false,"prefix":"","firstName":"Linkui","middleName":"","lastName":"Gao","suffix":""},{"id":585723643,"identity":"31991d20-90fc-4a6a-b000-405d40c26bc1","order_by":1,"name":"Gang Liu","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA2UlEQVRIiWNgGAWjYBACPmY4k4fxAYMBEVrY4FrYeJgNiNOCYPGwSRDlMDZ2HuMXP3fURvPP7z1W+aPgjjwD+9kDBBzGY2bZe+Z47oxjfGm3eQyeGTbw5CUQ1GLA23Yst+EYj9ltBoPDjA0SPPh9BNJi+BeoZT5QS+EPg8P2xGgxfszbVpO7AaiFgcfgcCIRWtjKmGXbDuRuPJZjLA3UktzGk4NfCz//4c0f37bV5c47fMbw448/h2372c8QjB1QdBxG4hJSDwTMHxgY6ohQNwpGwSgYBSMWAACQIz/skNXKJgAAAABJRU5ErkJggg==","orcid":"","institution":"Hengyang Normal University","correspondingAuthor":true,"prefix":"","firstName":"Gang","middleName":"","lastName":"Liu","suffix":""}],"badges":[],"createdAt":"2026-01-26 08:24:13","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-8698044/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-8698044/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":102060777,"identity":"74c0b547-902c-407f-a3be-079855d01a62","added_by":"auto","created_at":"2026-02-06 16:55:52","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":329981,"visible":true,"origin":"","legend":"","description":"","filename":"GL2026manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-8698044/v1_covered_e0ee6a34-0612-49eb-9a70-60e5176be66a.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"On the number of zeros of harmonic polynomials arranged in alternating patterns","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":true,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":true,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"analysis-and-mathematical-physics","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"aamp","sideBox":"Learn more about [Analysis and Mathematical Physics](http://link.springer.com/journal/13324)","snPcode":"13324","submissionUrl":"https://submission.nature.com/new-submission/13324/3","title":"Analysis and Mathematical Physics","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"em","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false},"keywords":"Harmonic polynomials, Argument Principle for Harmonic Functions, Descartes’ Rule of Signs, Cardano’s Formula","lastPublishedDoi":"10.21203/rs.3.rs-8698044/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-8698044/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"In this paper, we investigate the number of zeros of harmonic polynomials of the form$$f(z)=a_{1}z^{n_{1}}+a_{2}\\overline{z}^{n_{2}}+a_{3}z^{n_{3}}+\\cdots+\\left(\\frac{1+(-1)^{k+1}}{2}a_{k}z^{n_{k}}+\\frac{1+(-1)^{k}}{2}a_{k}\\overline{z}^{n_{k}}\\right)+a_{k+1},$$where $a_{j}~(j=1,2\\cdots,k+1)$ are non-zero constants and $n_{j}~(j=1,2\\cdots,k)$ are positive integers with $n_{1}\u003en_{2}\u003e\\cdots\u003en_{k}\\geq1$.Assume that $f$ is regular. 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