Combinatorial commutative algebra is a relatively new, rapidly developing

mathematical Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

discipline. As the name implies, it lies at the intersection of two more established fields,

commutative algebra Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideal (ring theory), ideals, and module (mathematics), modules over such rings. Both algebraic geometry and algebraic number theo ...

and

combinatorics Combinatorics is an area of mathematics primarily concerned with counting, both as a means and as an end to obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many ...

, and frequently uses methods of one to address problems arising in the other. Less obviously, polyhedral geometry plays a significant role. One of the milestones in the development of the subject was Richard Stanley's 1975

proof Proof most often refers to: * Proof (truth), argument or sufficient evidence for the truth of a proposition * Alcohol proof, a measure of an alcoholic drink's strength Proof may also refer to: Mathematics and formal logic * Formal proof, a co ...

of the Upper Bound Conjecture for

simplicial sphere In geometry and combinatorics, a simplicial (or combinatorial) ''d''-sphere is a simplicial complex homeomorphic to the ''d''-dimensional sphere. Some simplicial spheres arise as the boundaries of convex polytopes, however, in higher dimensions mos ...

s, which was based on earlier work of

Melvin Hochster Melvin Hochster (born August 2, 1943) is an American mathematician working in commutative algebra. He is currently the Jack E. McLaughlin Distinguished University Professor Emeritus of Mathematics at the University of Michigan. Education Hochs ...

and Gerald Reisner. While the problem can be formulated purely in geometric terms, the methods of the proof drew on commutative algebra techniques. A signature

theorem In mathematics and formal logic, a theorem is a statement (logic), statement that has been Mathematical proof, proven, or can be proven. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to esta ...

in combinatorial commutative algebra is the characterization of ''h''-vectors of

simplicial polytope In geometry, a simplicial polytope is a polytope whose facet_(mathematics), facets are all Simplex, simplices. For example, a ''simplicial polyhedron'' in three dimensions contains only Triangle, triangular faces conjecture In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis or Fermat's conjecture (now a theorem, proven in 1995 by Andrew Wiles), ha ...

d in 1970 by

Peter McMullen Peter McMullen (born 11 May 1942) is a British mathematician, a professor emeritus of mathematics at University College London. Education and career McMullen earned bachelor's and master's degrees from Trinity College, Cambridge, and studied at ...

. Known as the ''g''-theorem, it was proved in 1979 by Stanley (

necessity Necessary or necessity may refer to: Concept of necessity * Need ** An action somebody may feel they must do ** An important task or essential thing to do at a particular time or by a particular moment * Necessary and sufficient condition, in l ...

of the conditions, algebraic argument) and by

Louis Billera Louis Joseph Billera is a Professor of Mathematics at Cornell University. Career Billera completed his B.S. at the Rensselaer Polytechnic Institute in 1964. He earned his Ph.D. from the Graduate Center of the City University of New York in 1968, ...

and Carl W. Lee ( sufficiency, combinatorial and geometric construction). A major open question was the extension of this characterization from simplicial polytopes to simplicial spheres, the ''g''-conjecture, which was resolved in 2018 by

Karim Adiprasito Karim Alexander Adiprasito (born in 1988) is a German mathematician working at the University of Copenhagen and the Hebrew University of Jerusalem who works in combinatorics. He completed his PhD in 2013 at Free University Berlin under the super ...

Important notions of combinatorial commutative algebra

* Square-free

monomial ideal In abstract algebra, a monomial ideal is an ideal generated by monomials in a multivariate polynomial ring over a field. Definitions and properties Let \mathbb be a field and R = \mathbb /math> be the polynomial ring over \mathbb with ''n'' inde ...

in a

polynomial ring In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring formed from the set of polynomials in one or more indeterminates (traditionally also called variables) with coefficients in another ring, ...

and

Stanley–Reisner ring In mathematics, a Stanley–Reisner ring, or face ring, is a quotient of a polynomial ring, polynomial algebra over a field (algebra), field by a square-free monomial ideal, monomial ideal (ring theory), ideal. Such ideals are described more geomet ...

of a

simplicial complex In mathematics, a simplicial complex is a structured Set (mathematics), set composed of Point (geometry), points, line segments, triangles, and their ''n''-dimensional counterparts, called Simplex, simplices, such that all the faces and intersec ...

. *

Cohen–Macaulay ring In mathematics, a Cohen–Macaulay ring is a commutative ring with some of the algebro-geometric properties of a smooth variety, such as local equidimensionality. Under mild assumptions, a local ring is Cohen–Macaulay exactly when it is a fin ...

s. * Monomial ring, closely related to an affine semigroup ring and to the

coordinate ring In algebraic geometry, an affine variety or affine algebraic variety is a certain kind of algebraic variety that can be described as a subset of an affine space. More formally, an affine algebraic set is the set of the common zeros over an algeb ...

of an

affine Affine may describe any of various topics concerned with connections or affinities. It may refer to: * Affine, a Affinity_(law)#Terminology, relative by marriage in law and anthropology * Affine cipher, a special case of the more general substi ...

toric variety In algebraic geometry, a toric variety or torus embedding is an algebraic variety containing an algebraic torus as an open dense subset, such that the action of the torus on itself extends to the whole variety. Some authors also require it to be ...

. * Algebra with a straightening law. There are several versions of those, including Hodge algebras of

Corrado de Concini Corrado de Concini (born 28 July 1949, in Rome) is an Italian mathematician and professor at the Sapienza University of Rome. He studies algebraic geometry, quantum groups, invariant theory, and mathematical physics. Life and work He was born ...

David Eisenbud David Eisenbud (born 8 April 1947 in New York City) is an American mathematician. He is a professor of mathematics at the University of California, Berkeley and former director of the then Mathematical Sciences Research Institute (MSRI), now k ...

, and

Claudio Procesi Claudio Procesi (born 31 March 1941 in Rome) is an Italian mathematician, known for works in algebra and representation theory. Career Procesi studied at the Sapienza University of Rome, where he received his degree (Laurea) in 1963. In 1966 he ...

References

A foundational paper on Stanley–Reisner complexes by one of the pioneers of the theory: * The first book is a classic (first edition published in 1983): * Very influential, and well written, textbook-monograph: * Additional reading: * * * * A recent addition to the growing literature in the field, contains exposition of current research topics: * * * {{cite book , first1=Jürgen , last1=Herzog , first2=Takayuki , last2=Hibi, first3=Hidefumi , last3=Oshugi , title=Binomial Ideals , publisher=Springer , series=Graduate Texts in Mathematics , volume=279 , date=2018 , isbn=978-3-319-95349-6 , zbl=1403.13004, url={{GBurl, https://books.google.com.tr/books?id=HsoCuAEACAAJ Commutative algebra Algebraic geometry Algebraic combinatorics

Important notions of combinatorial commutative algebra

See also

References