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This work provides specific vanishing polynomials for the ring Z_m^×Z_l[x_1,x_2,…,x_n] under the condition (m,l)≠1, together with an explicit minimal strong Gröbner basis for the vanishing ideal of the ring Z_2^×Z_2[x]. The development uses a fully combinatorial proof strategy, aiming to make the algebraic structure of vanishing ideals explicit and computable.",{"@graph":63,"@context":118},[64,80,101],{"@type":65,"itemListElement":66},"BreadcrumbList",[67,71,74,77],{"item":68,"name":69,"@type":70,"position":9},"https://docshare.wps.com","Home","ListItem",{"item":72,"name":10,"@type":70,"position":73},"https://docshare.wps.com/template/",2,{"item":75,"name":51,"@type":70,"position":76},"https://docshare.wps.com/template/general/",3,{"item":78,"name":59,"@type":70,"position":79},"https://docshare.wps.com/template/explicit-grobner-basis-of-the-ideal-of-vanishing-polynomials/160957/",4,{"url":78,"name":59,"@type":81,"image":82,"author":87,"headline":59,"publisher":90,"fileFormat":93,"inLanguage":57,"description":61,"dateModified":94,"datePublished":95,"encodingFormat":93,"isAccessibleForFree":96,"interactionStatistic":97},"DigitalDocument",{"url":83,"@type":84,"width":85,"height":86},"https://docshare.wps.com/thumbnails/explicit-grobner-basis-of-the-ideal-of-vanishing-polynomials/160957.png","ImageObject",442,249,{"name":88,"@type":89},"Theodora","Person",{"url":68,"name":91,"@type":92},"DocShare","Organization","application/vnd.openxmlformats-officedocument.wordprocessingml.document","2026-09-20","2026-08-30",true,{"@type":98,"interactionType":99,"userInteractionCount":9},"InteractionCounter",{"@type":100},"ViewAction",{"@type":102,"mainEntity":103},"FAQPage",[104,110,114],{"name":105,"@type":106,"acceptedAnswer":107},"What is the vanishing ideal studied in the paper?","Question",{"text":108,"@type":109},"Vanishing polynomials form an ideal in a polynomial ring; this ideal is called the vanishing ideal. It collects all polynomials that evaluate to zero on every point of R^n.","Answer",{"name":111,"@type":106,"acceptedAnswer":112},"Which rings and conditions are considered for constructing vanishing polynomials?",{"text":113,"@type":109},"The paper considers vanishing polynomials in Z_m^×Z_l[x_1,x_2,…,x_n] with (m,l)≠1 and focuses on obtaining an explicit minimal strong Gröbner basis for the case Z_2^×Z_2[x].",{"name":115,"@type":106,"acceptedAnswer":116},"How does the paper characterize a minimal strong Gröbner basis?",{"text":117,"@type":109},"It defines strong Gröbner bases via divisibility properties of leading terms across all polynomials in the ideal, and calls the basis minimal when leading coefficients are 1 and leading power products are not divisibility-related for distinct basis elements.","https://schema.org",{"og:url":78,"og:type":120,"og:title":59,"og:site_name":91,"og:description":61},"article",{"robots":122,"canonical":78},"index,follow",{"doc_id":124,"site_id":56},160957,1788086450,{"code":4,"msg":5,"data":127},{"doc_id":124,"user_id":128,"nickname":88,"user_avatar":129,"doc_module":9,"category_id":50,"category_name":51,"doc_title":59,"doc_description":61,"doc_content":130,"file_id":131,"file_url":132,"file_type":133,"file_size":134,"view_count":9,"is_deleted":4,"is_public":9,"is_downloadable":9,"audit_status":9,"page_count":79,"language":135,"language_code":57,"site_id":56,"html_lang":57,"table_of_contents":136,"faqs":137,"seo_title":138,"seo_description":61,"update_tm":125,"read_time":73},687197207919,"https://ap-avatar.wpscdn.com/avatar/a000253d6f5f7c60be?x-image-process=image/resize,m_fixed,w_180,h_180&k=1779446848396160552","Explicit Gröbner Basis of the Ideal of Vanishing Polynomials\nAbdullah ÇAĞMAN1*, Abdullah ÇAĞMAN2 and Abdullah ÇAĞMAN3\n1 Ağrı İbrahim Çeçen University, Faculty of Science and Letters, Department of Mathematics, Ağrı, Turkey,\n\u0013 HYPERLINK \"mailto:acagman@agri.edu.tr\" \u0014acagman@agri.edu.tr\u0015\n2 Ağrı İbrahim Çeçen University, Faculty of Science and Letters, Department of Mathematics, Ağrı, Turkey,\n\u0013 HYPERLINK \"mailto:acagman@agri.edu.tr\" \u0014acagman@agri.edu.tr\u0015\n3 Ağrı İbrahim Çeçen University, Faculty of Science and Letters, Department of Mathematics, Ağrı, Turkey,\n\u0013 HYPERLINK \"mailto:acagman@agri.edu.tr\" \u0014acagman@agri.edu.tr\u0015\nAbstract\nVanishing polynomials form an ideal of polynomial ring over the coefficient ring. In this paper, we give some vanishing polynomials of the polynomial ring \u001aℤ\u001b𝑚\u001b×\u001aℤ\u001b𝑙\u001b[\u001a𝑥\u001b1\u001b,\u001a𝑥\u001b2\u001b,…,\u001a𝑥\u001b𝑛\u001b] where (𝑚,𝑙)≠1 and an explicit minimal strong Gröbner basis of the ideal of vanishing polynomials of the ring \u001aℤ\u001b2\u001b×\u001aℤ\u001b2\u001b[𝑥]. Our proof is based fully on a combinatorial way.\nKeywords: Vanishing polynomial, Polynomial ring, Vanishing Ideal, Gröbner Basis.\nIntroduction\nAlthough the notion of Gröbner basis firstly handled with the current name by Buchberger in his PhD thesis (Buchberger 1965), in 1927, Macaulay had been used this idea in his famous paper (Macaulay 1927).\nWhen Buchberger was studying Gröbner basis for polynomial rings in his thesis at the same time Hironoka suggested the standart basis idea which is a remarkable method for solving the important problem of algebraic geometry “resolution of singularities of algebraic varieties” (Hironaka 1964).\nReceived:\nRevised:\nAccepted:\n*Corresponding author: Abdullah ÇAĞMAN, PhD\nAgri Ibrahim Cecen University, Faculty of Science and Letters, Department of  Mathematics, Agrı Turkey\nE-mail: acagman@agri.edu.tr\nCite this article as: A. Çağman, A. Çağman and A. Çağman, Explicit Gröbner Basis of the Ideal of Vanishing Polynomials, Eastern Anatolian Journal of Science, Vol. 5, Issue 1, 1-4, 2019\nThough the statements Gröbner basis and standart basis are actually express the same things, Gröbner basis is come into prominence for the contribution to the computer algebra.\nAfter the efforts of Buchberger and Hironoka in the 1960s, Gröbner basis did not see sufficiently interest about twenty years. But, in the middle of 1980s, the software Macaulay designed by David Bayer and Michael Stilman became a groundbreaking development for the Gröbner basis studies. This computer algebra system is still in progress and used in several works about the Gröbner basis (Grayson and Stilman 2016).\nThe concept of Gröbner basis has been had many applications in different areas of mathematics. Solution of integer programming problem (Conti and Traverso 1991), graph theory (de Loera 1995), toric ideal theory serving as a bridge between the monomial ideal theory and the theory of triangulations of convex polytopes (Sturmfels 1996) and finding Markov basis in a statistical model (Aoki et al. 2010) are some examples of these applications.\nIn general, the interests of Gröbner basis are centered upon the field as a coefficient ring. The main reason for this is applications of Gröbner basis for arbitrary rings are very constraint. But, in recent years, several works with different coefficient rings have accelerated. For example, when proving correctness of data paths in system on chip design, usage of Gröbner basis in polynomial rings over Z_n has led to emergence of many works in this direction (Greuel et al. 2011), (Greuel et al. 2008), (Shekhar et al. 2005), (Wienand et al. 2008).\nLet R be an arbitrary coefficient ring with finite elements. The polynomial p∈R[x_1,x_2,...,x_n] is called vanishing if the image of all elements of R^nis zero under this polynomial. All vanishing polynomials form an ideal I which is called vanishing ideal. In this paper, our aim is to present an explicit Gröbner basis for the vanishing ideal I⊂Z_2×Z_2 [x].\nThis paper is organized as follows: Section 2 is devoted to some necessar","cbCaijM0nAvsZ132","https://ap.wps.com/l/cbCaijM0nAvsZ132","docx",49985,"English","# Abstract\n# Introduction\n# Materials and Methods\n## Gröbner basis and strong Gröbner basis\n# Results","[{\"question\":\"What is the vanishing ideal studied in the paper?\",\"answer\":\"Vanishing polynomials form an ideal in a polynomial ring; this ideal is called the vanishing ideal. It collects all polynomials that evaluate to zero on every point of R^n.\"},{\"question\":\"Which rings and conditions are considered for constructing vanishing polynomials?\",\"answer\":\"The paper considers vanishing polynomials in Z_m^×Z_l[x_1,x_2,…,x_n] with (m,l)≠1 and focuses on obtaining an explicit minimal strong Gröbner basis for the case Z_2^×Z_2[x].\"},{\"question\":\"How does the paper characterize a minimal strong Gröbner basis?\",\"answer\":\"It defines strong Gröbner bases via divisibility properties of leading terms across all polynomials in the ideal, and calls the basis minimal when leading coefficients are 1 and leading power products are not divisibility-related for distinct basis elements.\"}]","Explicit Gröbner Basis of the Ideal of Vanishing Polynomials | DOCX"]