[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-82231-en":3,"doc-seo-82231-105":28,"detail-sidebar-cat-0-en-105":90},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":11,"language":21,"language_code":22,"site_id":23,"html_lang":22,"table_of_contents":24,"faqs":25,"seo_title":13,"seo_description":14,"update_tm":26,"read_time":27},82231,962075114765,"Quinn","https://ap-avatar.wpscdn.com/davatar_a8503ba1806abce46bf441b54a3ca4cd",8,"Research & Report","Gårding’s Theorem for Posynomials","Extends Gårding’s theorem from homogeneous hyperbolic polynomials to homogeneous posynomials, where a finite positive sum of monomials with nonnegative real exponents is zero-free on a product of right half-planes. The degree-normalized root becomes concave, yielding a strengthened sector-to-fractional log-concavity implication: zero-freeness on a sector of aperture of form νπ leads to ν-fractional log-concavity. The sharper bound improves mixing-time and domain-sparsification guarantees for fixed-size matchings and nonsymmetric determinantal point processes.","arXiv :2607 .09 168v 1 [ cs .DS] 10 Jul 2026  \nGårding’s Theorem for Posynomials  \nNima Anari1  \n1 Stanford University, [anari@stanford.edu](anari@stanford.edu)  \nAbstract  \nWe extend Gårding’s theorem to homogeneous posynomials: if a finite positive sum of monomials with arbitrary nonnegative real exponents is zero-free on a product of right halfplanes, then its degree-normalized root is concave. Consequently, zero-freeness in a sector of aperture 􀀍􀀛 implies 􀀍-fractional log-concavity. This sharpens generic mixing and domainsparsification guarantees for fixed-size matchings and nonsymmetric determinantal point processes. The result was developed in an AI-assisted interaction initiated and checked by the author; Codex also assisted with assembling and typesetting the manuscript.  \n1 Introduction  \nGårding’s theorem turns a complex-analytic hypothesis into a convexity statement: the degreenormalized root of a homogeneous hyperbolic polynomial is concave on its hyperbolicity cone [Går59] . For a homogeneous polynomial with nonnegative coefficients, zero-freeness on a product of right half-planes therefore implies log-concavity on the positive orthant. This connection is now a standard tool in the study of negative dependence, log-concave generating polynomials, and Markov-chain sampling [BBL09; AOV21; BH20] .  \nWe prove the same theorem for posynomials. A posynomial is a finite sum  \n􀀽  \n􀀿 (􀁇) = Õ 􀀲 􀀰􀁇􀀰 = Õ 􀀲 􀀰 Ö 􀁇 􀀸 , 􀀲 􀀰 > 0, 􀀰 ∈ ℝ0 .  \n􀀰∈􀁁 􀀰∈􀁁 􀀸 =1  \nIt is homogeneous of degree 􀀳 if Í􀀸 􀀰 􀀸 = 􀀳 for every 􀀰 ∈ 􀁁 . On ℂ \\ (−∞, 0], real powers are defined using the principal logarithm; this convention defines a holomorphic branch of 􀀿 on the corresponding product domain.  \nFor 0 \u003C 􀀍 ≤ 1, let  \nΓ􀀍 = {􀁁􀀴 􀀸􀀔 | 􀁁 > 0, |􀀔| \u003C 􀀍􀀛/2} .  \nThus Γ1 is the open right half-plane. We say that 􀀿 is Γ􀀍 -stable if its principal branch has no zero on Γ􀀽􀀍 . We use arg(􀁉) ∈ (−􀀛 , 􀀛 ) for the principal argument; when a zero-free path specifies a continuous lift, we write Arg instead.  \nTheorem 1 (Gårding’s theorem for posynomials). Let 􀀿 be a homogeneous posynomial of degree 􀀳 > 0. If 􀀿 is Γ1-stable, then  \n􀁇 ↦−→ 􀀿 (􀁇)1/􀀳  \nis concave on ℝ0 . In particular, log 􀀿 is concave on ℝ0 .  \nThe half-plane statement contains the sector statement by a power map.  \nCorollary 2 (Sharp fractional log-concavity). Let 0 \u003C 􀀍 ≤ 1, and let 􀀿 be a homogeneous posynomial of degree 􀀳 > 0. If 􀀿 is Γ􀀍 -stable, then  \n􀁇 ↦−→ 􀀿 (􀁇 􀀍 )1/(􀀍 􀀳)  \nis concave on ℝ0 . Consequently, 􀁇 →↦ log 􀀿(􀁇 􀀍 ) is concave.  \nThe dependence on 􀀍 is best possible. Indeed, 􀀿(􀁇, 􀁈) = 􀁇 􀀎 + 􀁈􀀎 is Γ􀀍 -stable exactly when 􀀍􀀎 ≤ 1, and (􀁇􀀍􀀎 + 􀁈􀀍􀀎 )1/(􀀍􀀎) is concave on the positive orthant exactly in the same range.  \nWe consider real exponents in order to sharpen the known connection between sector-stability and fractional log-concavity. A homogeneous polynomial 􀀵 is 􀀍-fractionally log-concave if  \n􀁇 ↦−→ log 􀀵 (􀁇 􀀍 )  \nis concave on the positive orthant. Earlier work showed that Γ2􀀍-stability implies 􀀍-fractional log-concavity [Ali+21, Lemma 69] . Thus a sector of aperture 􀀎􀀛 yielded only 􀀎/2-fractional log-concavity. Corollary 2 removes this factor-of-two loss.  \nThis loss propagated into the polynomial exponents in subsequent sampling results. Removing it yields quadratic improvements in the resulting mixing-time and domain-sparsification guarantees, including those for fixed-size matchings and nonsymmetric determinantal point processes [Ana+21; Ana+22] .  \n2 Ray Monotonicity for Real Exponents  \nWe first record the real-exponent form of Descartes’ rule. For a finite sequence of nonzero real numbers 􀀱0 , . . . , 􀀱 􀀣, let 􀀫(􀀱0 , . . . , 􀀱 􀀣) denote its number of sign changes.  \nLemma 3 (Generalized Descartes rule). Let 􀀗0 \u003C · · · \u003C 􀀗 􀀣 be real numbers and let  \n􀀣  \n􀀛 (􀁃) = Õ 􀀱 􀀹􀁃 􀀗 􀀹 , 􀀱 􀀹 ∈ ℝ \\ {0} .  \n􀀹 =0  \nThe number of zeros of 􀀛 on ℝ >0 , counted with multiplicity, is at most 􀀫 (􀀱0 , . . . , 􀀱 􀀣).  \nProof. We argue by induction on the number of terms. Multiplication by 􀁃−􀀗0 does not change the po","cbCaid03v337H7lg","https://ap.wps.com/l/cbCaid03v337H7lg","pdf",201356,1,"English","en",105,"# Abstract\n# Introduction\n## Posynomials and Γν-stability\n## Theorem 1: Gårding’s theorem for posynomials\n## Corollary 2: Sharp fractional log-concavity\n# Ray Monotonicity for Real Exponents\n## Lemma 3: Generalized Descartes rule\n## Theorem 4: Sendov–Sendov ray monotonicity","[{\"question\":\"What does the main result say about homogeneous posynomials that are Γ1-stable?\",\"answer\":\"If a homogeneous posynomial of degree d\\u003e0 is Γ1-stable (zero-free on the corresponding right half-plane domain), then the map x↦p(x)^(1/d) is concave on the nonnegative orthant, and log p is concave there.\"},{\"question\":\"How does sector zero-freeness relate to fractional log-concavity?\",\"answer\":\"Zero-freeness on a sector of aperture νπ implies ν-fractional log-concavity. The document shows that the induced concavity can be sharpened without the earlier factor-of-two loss.\"},{\"question\":\"Why are the improved bounds important for sampling problems?\",\"answer\":\"The strengthened concavity translates into quadratic improvements in mixing-time and domain-sparsification guarantees for fixed-size matchings and nonsymmetric determinantal point processes.\"}]",1784179003,20,{"code":4,"msg":29,"data":30},"ok",{"site_id":23,"language":22,"slug":31,"title":13,"keywords":32,"description":14,"schema_data":33,"social_meta":85,"head_meta":87,"extra_data":89,"updated_unix":26},"gardings-theorem-for-posynomials","",{"@graph":34,"@context":84},[35,52,67],{"@type":36,"itemListElement":37},"BreadcrumbList",[38,42,46,49],{"item":39,"name":40,"@type":41,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":43,"name":44,"@type":41,"position":45},"https://docshare.wps.com/document/","Document",2,{"item":47,"name":12,"@type":41,"position":48},"https://docshare.wps.com/document/research-report/",3,{"item":50,"name":13,"@type":41,"position":51},"https://docshare.wps.com/document/gardings-theorem-for-posynomials/82231/",4,{"url":50,"name":13,"@type":53,"author":54,"headline":13,"publisher":56,"fileFormat":59,"inLanguage":22,"description":14,"dateModified":60,"datePublished":61,"encodingFormat":59,"isAccessibleForFree":62,"interactionStatistic":63},"DigitalDocument",{"name":9,"@type":55},"Person",{"url":39,"name":57,"@type":58},"DocShare","Organization","application/pdf","2026-07-17","2026-07-16",true,{"@type":64,"interactionType":65,"userInteractionCount":20},"InteractionCounter",{"@type":66},"ViewAction",{"@type":68,"mainEntity":69},"FAQPage",[70,76,80],{"name":71,"@type":72,"acceptedAnswer":73},"What does the main result say about homogeneous posynomials that are Γ1-stable?","Question",{"text":74,"@type":75},"If a homogeneous posynomial of degree d>0 is Γ1-stable (zero-free on the corresponding right half-plane domain), then the map x↦p(x)^(1/d) is concave on the nonnegative orthant, and log p is concave there.","Answer",{"name":77,"@type":72,"acceptedAnswer":78},"How does sector zero-freeness relate to fractional log-concavity?",{"text":79,"@type":75},"Zero-freeness on a sector of aperture νπ implies ν-fractional log-concavity. The document shows that the induced concavity can be sharpened without the earlier factor-of-two loss.",{"name":81,"@type":72,"acceptedAnswer":82},"Why are the improved bounds important for sampling problems?",{"text":83,"@type":75},"The strengthened concavity translates into quadratic improvements in mixing-time and domain-sparsification guarantees for fixed-size matchings and nonsymmetric determinantal point processes.","https://schema.org",{"og:url":50,"og:type":86,"og:title":13,"og:site_name":57,"og:description":14},"article",{"robots":88,"canonical":50},"index,follow",{"doc_id":7,"site_id":23},{"code":4,"msg":5,"data":91},[92,96,100,104,109,114,119,122,126,129,133],{"id":20,"doc_module":4,"doc_module_name":44,"category_name":93,"show_sort_weight":94,"slug":95},"Story & Novel",90,"story-novel",{"id":45,"doc_module":4,"doc_module_name":44,"category_name":97,"show_sort_weight":98,"slug":99},"Literature",80,"literature",{"id":51,"doc_module":4,"doc_module_name":44,"category_name":101,"show_sort_weight":102,"slug":103},"Exam",70,"exam",{"id":105,"doc_module":4,"doc_module_name":44,"category_name":106,"show_sort_weight":107,"slug":108},5,"Comic",60,"comic",{"id":110,"doc_module":4,"doc_module_name":44,"category_name":111,"show_sort_weight":112,"slug":113},6,"Technology",50,"technology",{"id":115,"doc_module":4,"doc_module_name":44,"category_name":116,"show_sort_weight":117,"slug":118},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":44,"category_name":12,"show_sort_weight":120,"slug":121},30,"research-report",{"id":123,"doc_module":4,"doc_module_name":44,"category_name":124,"show_sort_weight":27,"slug":125},9,"Religion & Spirituality","religion-spirituality",{"id":27,"doc_module":4,"doc_module_name":44,"category_name":127,"show_sort_weight":27,"slug":128},"World Cup","world-cup",{"id":130,"doc_module":4,"doc_module_name":44,"category_name":131,"show_sort_weight":130,"slug":132},10,"Lifestyle","lifestyle",{"id":134,"doc_module":4,"doc_module_name":44,"category_name":135,"show_sort_weight":105,"slug":136},19,"General","general"]