Combinatorial Nullstellensatz

E621146

result in algebraic combinatorics theorem

Combinatorial Nullstellensatz is a powerful algebraic tool in combinatorics that uses polynomial methods over fields to derive results about combinatorial structures, such as existence and counting theorems.

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Statements (47)

Predicate	Object
instanceOf	result in algebraic combinatorics ⓘ theorem ⓘ
appliesTo	arbitrary fields ⓘ finite fields ⓘ
assumes	finite degree of polynomials ⓘ specified leading monomial with nonzero coefficient ⓘ
concerns	Cartesian products of subsets of a field ⓘ coefficients of monomials ⓘ degrees of polynomials ⓘ multivariate polynomials ⓘ
conclusion	if the coefficient of a certain monomial is nonzero then the polynomial does not vanish identically on a given grid ⓘ
field	algebra ⓘ combinatorics ⓘ
guarantees	existence of a nonvanishing evaluation of a polynomial on a grid under degree conditions ⓘ
hasProofTechnique	Lagrange interpolation NERFINISHED ⓘ algebraic manipulation of coefficients ⓘ induction on degree ⓘ
hasVariant	coefficient formula version ⓘ multicolored version ⓘ nonvanishing version ⓘ
implies	counting results in combinatorics ⓘ existence results in combinatorics ⓘ
influenced	development of algebraic methods in combinatorics ⓘ
introducedBy	Noga Alon NERFINISHED ⓘ
mainIdea	relates coefficients of multivariate polynomials to evaluations on Cartesian products of subsets of a field ⓘ
publicationYear	1999 ⓘ
publishedIn	Journal of Combinatorial Theory Series A NERFINISHED ⓘ
relatedTo	Alon–Tarsi conjecture NERFINISHED ⓘ Chevalley–Warning theorem NERFINISHED ⓘ Erdos–Heilbronn conjecture NERFINISHED ⓘ polynomial method in additive combinatorics ⓘ
statedOver	commutative field ⓘ
taughtIn	courses on polynomial methods ⓘ graduate courses in combinatorics ⓘ
toolFor	bounding sizes of combinatorial configurations ⓘ coloring problems in graphs and hypergraphs ⓘ establishing existence of transversals ⓘ proving combinatorial identities ⓘ
usedFor	Erdos–Ko–Rado type problems NERFINISHED ⓘ additive combinatorics ⓘ design theory ⓘ graph theory ⓘ number theory ⓘ polynomial method in combinatorics ⓘ zero-sum problems ⓘ
uses	polynomial method ⓘ polynomials over fields ⓘ

Referenced by (1)

Full triples — surface form annotated when it differs from this entity's canonical label.

Noga Alon → notableWork → Combinatorial Nullstellensatz ⓘ