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Polynomials & Algebraic Invariants

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Abstractions about polynomial-like algebraic objects and their invariants, covering generalized polynomial forms (Laurent polynomials, quasi-polynomials, exponential polynomials), invariant-theoretic constructions (invariant polynomial, characteristic polynomial of a graph, Contou-Carrère symbol), and algebraic-geometric polynomial structures (determinantal variety, generic fiber, Rees decomposition).

20 abstractions in this family — domain-specific abstractions that sit near one another in structural-signature space (k-means over structural-signature embeddings). Each is shown with its short description.

  • Algebraic normal form — The unique multilinear polynomial over GF(2) representing a Boolean function as an XOR of square-free AND monomials and an optional constant.
  • Analytic Function — A real or complex function that, around every point of its open domain, equals a convergent power series in the relevant coordinates.
  • Characteristic polynomial of a graph — In spectral graph theory, the characteristic polynomial of a graph is the characteristic polynomial of its adjacency matrix.
  • Contou-Carrère symbol — A multiplicative Steinberg symbol on pairs of invertible Laurent series over an Artinian ring, valued in the ring's units and defined through winding exponents, leading coefficients, and positive-negative coefficient pairings.
  • Determinantal variety — The radical ideal defining the determinantal variety is generated by the (r + 1) × (r + 1) minors of the matrix (Bruns-Vetter, Theorem 2.10).
  • Euler's elliptic differential equation — A branch-qualified relation between two elliptic differentials sharing a nonsingular quartic, whose local integration yields an algebraic addition relation.
  • Exponential polynomial — In mathematics, exponential polynomials are functions on fields, rings, or abelian groups that take the form of polynomials in a variable and an exponential function.
  • Generic fiber — The generic fiber of a morphism of schemes is the fiber obtained over the generic point of the base, capturing the behavior valid on a dense open part of that base.
  • Hermite spline — In the mathematical subfield of numerical analysis, a Hermite spline is a spline curve where each polynomial of the spline is in Hermite form.
  • Invariant factorization of LPDOs — Each Laplace invariant is an explicit polynomial condition of factorization; coefficients of this polynomial are explicit functions of the coefficients of the initial LPDO.
  • Invariant polynomial — In mathematics, an invariant polynomial is a polynomial P that is invariant under a group \Gamma acting on a vector space V .
  • Laurent Polynomial — A Laurent polynomial is a finite linear combination of integer powers of one or more variables, allowing negative as well as nonnegative exponents.
  • Maclaurin's Inequality — The descending chain of root-normalized elementary symmetric means of a finite nonnegative real vector.
  • Polynomial — A finite formal sum of monomials with coefficients in a declared ring and nonnegative integer exponents, distinguished from the function obtained by evaluating it.
  • Polynomial Content and Primitive Part — Separate a nonzero polynomial over a unique factorization domain into its coefficient gcd and a residual polynomial with unit coefficient gcd.
  • Pseudorandom generators for polynomials — Pseudorandom generators for low-degree polynomials are a particular instance of pseudorandom generators for statistical tests, where the statistical tests considered are evaluations of low-degree polynomials.
  • Quasi-polynomial — In mathematics, a quasi-polynomial (sometimes called pseudo-polynomial) is a generalization of polynomials.
  • Rainville polynomials — A polynomial family defined by the generating function involving the modified Bessel function I0.
  • Rees decomposition — In commutative algebra, a Rees decomposition is a way of writing a ring in terms of polynomial subrings.
  • Restricted Power Series — A formal power series over a complete linearly topologized ring whose coefficients tend to zero in the ring topology as degree grows, equivalently an element of the completed polynomial ring.