TY - JOUR

T1 - Subalgebras in K[x] of small codimension

AU - Grönkvist, Rode

AU - Leffler, Erik

AU - Torstensson, Anna

AU - Ufnarovski, Victor

PY - 2022

Y1 - 2022

N2 - We introduce the concept of subalgebra spectrum, Sp(A), for a subalgebra A of finite codimension in K[x]. The spectrum is a finite subset of the underlying field. We also introduce a tool, the characteristic polynomial of A, which has the spectrum as its set of zeroes. The characteristic polynomial can be computed from the generators of A, thus allowing us to find the spectrum of an algebra given by generators. We proceed by using the spectrum to get descriptions of subalgebras of finite codimension. More precisely we show that A can be described by a set of conditions that each is either of the type f(α) = f(β) for α, β in Sp(A) or of the type stating that some linear combination of derivatives of different orders evaluated in elements of Sp(A) equals zero. We use these types of conditions to, by an inductive process, find explicit descriptions of subalgebras of codimension up to three. These descriptions also include SAGBI bases for each family of subalgebras.

AB - We introduce the concept of subalgebra spectrum, Sp(A), for a subalgebra A of finite codimension in K[x]. The spectrum is a finite subset of the underlying field. We also introduce a tool, the characteristic polynomial of A, which has the spectrum as its set of zeroes. The characteristic polynomial can be computed from the generators of A, thus allowing us to find the spectrum of an algebra given by generators. We proceed by using the spectrum to get descriptions of subalgebras of finite codimension. More precisely we show that A can be described by a set of conditions that each is either of the type f(α) = f(β) for α, β in Sp(A) or of the type stating that some linear combination of derivatives of different orders evaluated in elements of Sp(A) equals zero. We use these types of conditions to, by an inductive process, find explicit descriptions of subalgebras of codimension up to three. These descriptions also include SAGBI bases for each family of subalgebras.

KW - Derivation

KW - Resultant

KW - SAGBI basis

KW - Subalgebra spectrum

U2 - 10.1007/s00200-022-00573-4

DO - 10.1007/s00200-022-00573-4

M3 - Article

AN - SCOPUS:85136576284

VL - 33

SP - 751

EP - 789

JO - Applicable Algebra in Engineering, Communications and Computing

JF - Applicable Algebra in Engineering, Communications and Computing

SN - 1432-0622

IS - 6

ER -