scholarly journals COMMUTATIVE LEIBNIZ-POISSON ALGEBRAS OF POLYNOMIAL GROWTH

2017 ◽  
Vol 18 (3.1) ◽  
pp. 54-65
Author(s):  
S.M. Ratseev
Keyword(s):  
Polynomial Growth ◽  
Finite Basis ◽  
Polynomial Identities ◽  
Poisson Algebras

In this paper we study commutative Leibniz-Poisson algebras. We prove that a variety of commutative Leibniz-Poisson algebras has either polynomial growth or growth with exponential not less than 2, the field being arbitrary. We prove that every variety of commutative Leibniz-Poisson algebras of polynomial growth over a field of characteristic 0 has a finite basis for its polynomial identities. Also we construct a variety of commutative Leibniz-Poisson algebras with almost polynomial growth.

Poisson algebras of polynomial growth

2013 ◽  
Vol 54 (3) ◽  
pp. 555-565 ◽  
Author(s):  
S. M. Ratseev
Keyword(s):  
Polynomial Growth ◽  
Poisson Algebras

Witnessing matrix identities and proof complexity

2018 ◽  
Vol 28 (02) ◽  
pp. 217-256
Author(s):  
Fu Li ◽  
Iddo Tzameret
Keyword(s):  
Computational Complexity ◽  
Lower Bound ◽  
Finite Basis ◽  
Proof Complexity ◽  
Polynomial Identities ◽  
Open Problems ◽  
Complete Proof ◽  
Proof Systems ◽  
Matrix Identity ◽  
Matrix Identities

We use results from the theory of algebras with polynomial identities (PI-algebras) to study the witness complexity of matrix identities. A matrix identity of [Formula: see text] matrices over a field [Formula: see text]is a non-commutative polynomial (f(x1, …, xn)) over [Formula: see text], such that [Formula: see text] vanishes on every [Formula: see text] matrix assignment to its variables. For every field [Formula: see text]of characteristic 0, every [Formula: see text] and every finite basis of [Formula: see text] matrix identities over [Formula: see text], we show there exists a family of matrix identities [Formula: see text], such that each [Formula: see text] has [Formula: see text] variables and requires at least [Formula: see text] many generators to generate, where the generators are substitution instances of elements from the basis. The lower bound argument uses fundamental results from PI-algebras together with a generalization of the arguments in [P. Hrubeš, How much commutativity is needed to prove polynomial identities? Electronic colloquium on computational complexity, ECCC, Report No.: TR11-088, June 2011].We apply this result in algebraic proof complexity, focusing on proof systems for polynomial identities (PI proofs) which operate with algebraic circuits and whose axioms are the polynomial-ring axioms [P. Hrubeš and I. Tzameret, The proof complexity of polynomial identities, in Proc. 24th Annual IEEE Conf. Computational Complexity, CCC 2009, 15–18 July 2009, Paris, France (2009), pp. 41–51; Short proofs for the determinant identities, SIAM J. Comput. 44(2) (2015) 340–383], and their subsystems. We identify a decrease in strength hierarchy of subsystems of PI proofs, in which the [Formula: see text]th level is a sound and complete proof system for proving [Formula: see text] matrix identities (over a given field). For each level [Formula: see text] in the hierarchy, we establish an [Formula: see text] lower bound on the number of proof-steps needed to prove certain identities.Finally, we present several concrete open problems about non-commutative algebraic circuits and speed-ups in proof complexity, whose solution would establish stronger size lower bounds on PI proofs of matrix identities, and beyond.

POLYNOMIAL IDENTITIES ON SUPERALGEBRAS AND ALMOST POLYNOMIAL GROWTH

2001 ◽  
Vol 29 (9) ◽  
pp. 3787-3800 ◽  
Author(s):  
A. Giambruno ◽  
S. Mishchenko ◽  
M. Zaicev
Keyword(s):  
Polynomial Growth ◽  
Polynomial Identities

scholarly journals Identities and central polynomials for real graded division algebras

2017 ◽  
Vol 27 (07) ◽  
pp. 935-952
Author(s):  
Diogo Diniz ◽  
Claudemir Fidelis ◽  
Sérgio Mota
Keyword(s):  
Abelian Group ◽  
Finite Abelian Group ◽  
Finite Basis ◽  
Division Algebras ◽  
Polynomial Identities ◽  
Finite Dimensional ◽  
Real Algebra ◽  
Graded Polynomial Identities

Let [Formula: see text] be a finite dimensional simple real algebra with a division grading by a finite abelian group [Formula: see text]. In this paper, we provide a finite basis for the [Formula: see text]-ideal of graded polynomial identities for [Formula: see text] and a finite basis for the [Formula: see text]-space of graded central polynomials for [Formula: see text].

scholarly journals Weak polynomial identities and their applications

2021 ◽  
Vol 29 (2) ◽  
pp. 291-324
Author(s):  
Vesselin Drensky
Keyword(s):  
Associative Algebra ◽  
Finite Basis ◽  
Vector Subspace ◽  
Free Associative Algebra ◽  
Polynomial Identities ◽  
Finite Basis Problem ◽  
Basis Problem ◽  
Nonassociative Algebras ◽  
Survey Results ◽  
Algebra Over A Field

Abstract Let R be an associative algebra over a field K generated by a vector subspace V. The polynomial f(x 1, . . . , xn ) of the free associative algebra K〈x 1, x 2, . . .〉 is a weak polynomial identity for the pair (R, V) if it vanishes in R when evaluated on V. We survey results on weak polynomial identities and on their applications to polynomial identities and central polynomials of associative and close to them nonassociative algebras and on the finite basis problem. We also present results on weak polynomial identities of degree three.

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