The Polynomial Semiring¶

The Polynomial Semiring \(R[x]\) provides a generic algebraic structure for working with univariate polynomials over a coefficient semiring \(R\). It is a specialized subclass of the Monoid Algebra Semiring where keys are non-negative integer exponents in \((\mathbb{N}_0, +)\).

It finds wide applications in discrete signal processing (as FIR filters), coding theory, and formal algebra.

Mathematical Definition¶

• Set (\(S\)): Univariate polynomials, represented as sparse mappings {exponent: coefficient}.
  • Example: The polynomial \(1 + 2x + 4x^2\) is represented as {0: 1, 1: 2, 2: 4}.
• Coefficient Semiring (\(R_C\)): An underlying semiring defining addition and multiplication for the coefficients.
• Addition (\(+\)): Element-wise addition of coefficients for matching exponents.
• Multiplication (\(\cdot\)): Standard polynomial multiplication, corresponding to the discrete convolution of coefficients.
• Additive Identity (\(0\)): The zero polynomial {}.
• Multiplicative Identity (\(1\)): {0: R_C.one} (constant polynomial 1).

Implementation in algebrax¶

PolynomialSemiring inherits from MonoidAlgebraSemiring[int, T].

Example: Polynomials over Integers¶

import algebrax as ax

# 1. Define the coefficient semiring (integers)
int_semiring = ax.semiring.StandardSemiring(int)

# 2. Initialize the ax.semiring.PolynomialSemiring (R[x])
poly_semiring = ax.semiring.PolynomialSemiring(int_semiring)

# p1(x) = 1 + 2x
p1 = {0: 1, 1: 2}

# p2(x) = 3 + 4x^2
p2 = {0: 3, 2: 4}

# --- Operations ---

# Addition: (1 + 2x) + (3 + 4x^2) = 4 + 2x + 4x^2
added = poly_semiring.add(p1, p2)
# Result: {0: 4, 1: 2, 2: 4}
print(f"p1 + p2 = {added}")

# Multiplication: (1 + 2x) * (3 + 4x^2) = 3 + 6x + 4x^2 + 8x^3
multiplied = poly_semiring.mul(p1, p2)
# Result: {0: 3, 1: 6, 2: 4, 3: 8}
print(f"p1 * p2 = {multiplied}")

Use Cases¶

• Signal Processing: Polynomial multiplication is equivalent to discrete signal convolution, used in Finite Impulse Response (FIR) filters.
• Error-Correcting Codes: Polynomials over finite fields are fundamental to codes like Reed-Solomon.
• Abstract Algebra: As a building block for formal power series, quotient rings, and field extensions.