🌿Computational Algebraic Geometry Unit 3 – Polynomials and Rings in Algebra

Key Concepts and Definitions

• Rings generalize the arithmetic of integers and polynomials consist of a set equipped with two binary operations (usually addition and multiplication) that satisfy certain axioms
• Polynomials are expressions consisting of variables and coefficients combined using addition, subtraction, and multiplication
• Monomials are the building blocks of polynomials consist of a single term, which is a product of a coefficient and one or more variables raised to non-negative integer exponents
• Degree of a polynomial refers to the highest degree among its monomials determined by summing the exponents of the variables in each monomial
• Leading coefficient of a polynomial is the coefficient of the monomial with the highest degree
• Zero polynomial is the polynomial where all coefficients are zero has a degree of negative infinity

Polynomial Basics

• Polynomials can be represented in standard form by arranging the terms in descending order of degree and combining like terms
• Addition and subtraction of polynomials involve combining like terms (monomials with the same variables and exponents)
• Multiplication of polynomials follows the distributive law and involves multiplying each term of one polynomial by each term of the other, then combining like terms
• Division of polynomials can be performed using long division or synthetic division
  • Long division involves dividing the highest degree term of the dividend by the highest degree term of the divisor and repeating the process with the remainder
  • Synthetic division is a shortcut method for dividing a polynomial by a linear factor $(x - a)$
• Evaluation of a polynomial at a specific value involves substituting the value for the variable and simplifying the resulting expression
• Polynomial functions are functions defined by polynomials, where the input is the variable and the output is the value of the polynomial at that input

Ring Theory Fundamentals

• Rings are algebraic structures that generalize the arithmetic of integers and polynomials
• A ring $(R, +, \cdot)$ consists of a set $R$ equipped with two binary operations, addition $(+)$ and multiplication $(\cdot)$, satisfying the following axioms:
  • $(R, +)$ is an abelian group (associativity, commutativity, identity element, inverses)
  • Multiplication is associative: $a \cdot (b \cdot c) = (a \cdot b) \cdot c$ for all $a, b, c \in R$
  • Distributive laws hold: $a \cdot (b + c) = a \cdot b + a \cdot c$ and $(a + b) \cdot c = a \cdot c + b \cdot c$ for all $a, b, c \in R$
• Commutative rings are rings where multiplication is commutative: $a \cdot b = b \cdot a$ for all $a, b \in R$
• Unity or identity element of a ring is an element $1 \in R$ such that $1 \cdot a = a \cdot 1 = a$ for all $a \in R$
• Zero element of a ring is an element $0 \in R$ such that $a + 0 = a$ for all $a \in R$ and $a \cdot 0 = 0 \cdot a = 0$ for all $a \in R$
• Integral domain is a commutative ring with unity and no zero divisors (if $a \cdot b = 0$, then either $a = 0$ or $b = 0$)

Polynomial Rings and Their Properties

• Polynomial rings are rings formed by the set of polynomials with coefficients from a given ring
• Given a ring $R$, the polynomial ring $R[x]$ consists of all polynomials with coefficients in $R$ and variable $x$
  • Elements of $R[x]$ are of the form $a_0 + a_1x + a_2x^2 + \cdots + a_nx^n$, where $a_i \in R$ and $n \geq 0$
• Addition and multiplication in polynomial rings are performed by treating the polynomials as formal expressions and applying the usual rules for polynomial arithmetic
• If $R$ is a commutative ring, then $R[x]$ is also a commutative ring
• If $R$ is an integral domain, then $R[x]$ is also an integral domain
• Degree of a polynomial in $R[x]$ is the highest power of $x$ appearing in the polynomial, with the convention that the zero polynomial has degree $-\infty$
• Polynomial rings can be extended to multiple variables, denoted as $R[x_1, x_2, \ldots, x_n]$, where the polynomials have coefficients in $R$ and variables $x_1, x_2, \ldots, x_n$

Operations on Polynomials

• Addition of polynomials in $R[x]$ is performed by adding the coefficients of like terms
  • $(a_0 + a_1x + \cdots + a_nx^n) + (b_0 + b_1x + \cdots + b_mx^m) = (a_0 + b_0) + (a_1 + b_1)x + \cdots + (a_k + b_k)x^k$, where $k = \max(n, m)$ and $a_i = 0$ for $i > n$ and $b_i = 0$ for $i > m$
• Subtraction of polynomials is similar to addition, with the coefficients of the second polynomial being subtracted from the corresponding coefficients of the first polynomial
• Multiplication of polynomials in $R[x]$ is performed by multiplying each term of one polynomial by each term of the other and then adding the resulting terms
  • $(a_0 + a_1x + \cdots + a_nx^n) \cdot (b_0 + b_1x + \cdots + b_mx^m) = \sum_{i=0}^n \sum_{j=0}^m a_ib_jx^{i+j}$
• Division of polynomials in $R[x]$ can be performed using long division or, in some cases, by factoring the polynomials
  • Long division of polynomials is similar to long division of integers, where the dividend is divided by the divisor, and the process is repeated with the remainder until the degree of the remainder is less than the degree of the divisor
  • If $R$ is a field, then polynomial division in $R[x]$ always results in a unique quotient and remainder

Factorization and Irreducibility

• Factorization of a polynomial is the process of expressing the polynomial as a product of lower-degree polynomials
• A polynomial $f(x) \in R[x]$ is reducible over $R$ if it can be expressed as the product of two non-constant polynomials in $R[x]$; otherwise, it is irreducible over $R$
• In an integral domain $R$, a polynomial $f(x) \in R[x]$ is prime if whenever $f(x)$ divides the product $g(x)h(x)$ for some $g(x), h(x) \in R[x]$, then $f(x)$ divides either $g(x)$ or $h(x)$
• Unique factorization domains (UFDs) are integral domains where every non-zero, non-unit element can be uniquely factored as a product of prime elements, up to the order and unit factors
  • Examples of UFDs include the integers $\mathbb{Z}$, polynomial rings over fields, and polynomial rings over UFDs
• Gauss's Lemma states that if $R$ is a UFD, then $R[x]$ is also a UFD
• Eisenstein's criterion is a sufficient condition for a polynomial with integer coefficients to be irreducible over the rational numbers

Applications in Algebraic Geometry

• Algebraic geometry studies geometric objects defined by polynomial equations
• Affine varieties are sets of points in affine space that satisfy a system of polynomial equations
  • For a field $k$ and polynomials $f_1, \ldots, f_s \in k[x_1, \ldots, x_n]$, the affine variety $V(f_1, \ldots, f_s)$ is the set of all points $(a_1, \ldots, a_n) \in k^n$ such that $f_i(a_1, \ldots, a_n) = 0$ for all $1 \leq i \leq s$
• Ideal of a variety is the set of all polynomials that vanish on the variety
  • For an affine variety $V \subseteq k^n$, the ideal $I(V)$ is the set $\{f \in k[x_1, \ldots, x_n] : f(a_1, \ldots, a_n) = 0 \text{ for all } (a_1, \ldots, a_n) \in V\}$
• Hilbert's Nullstellensatz (Theorem of Zeros) relates ideals and varieties over algebraically closed fields
  • For an algebraically closed field $k$ and an ideal $I \subseteq k[x_1, \ldots, x_n]$, the ideal of the variety of $I$ is equal to the radical of $I$: $I(V(I)) = \sqrt{I}$
• Groebner bases are a powerful tool in computational algebraic geometry for solving systems of polynomial equations and ideal membership problems
  • A Groebner basis is a particular generating set of an ideal with nice properties that allow for efficient computation

Advanced Topics and Extensions

• Polynomial rings over non-commutative rings, such as matrix rings or quaternions, lead to non-commutative polynomial rings with interesting properties
• Skew polynomial rings are a generalization of polynomial rings where the coefficients come from a non-commutative ring, and the multiplication of variables and coefficients is subject to a skew-commutation rule
• Laurent polynomial rings are a generalization of polynomial rings that allow for negative exponents of variables
  • The Laurent polynomial ring over a ring $R$ is denoted $R[x, x^{-1}]$ and consists of elements of the form $\sum_{i=k}^n a_ix^i$, where $k, n \in \mathbb{Z}$ with $k \leq n$ and $a_i \in R$
• Polynomial rings over rings with additional structure, such as graded rings or topological rings, inherit some of the properties of the coefficient ring
• Algebraic geometry can be extended to schemes, which are a generalization of varieties that allow for more general spaces and provide a unified framework for studying both affine and projective varieties
• Computational methods, such as Buchberger's algorithm for computing Groebner bases and the F4/F5 algorithms for solving polynomial systems, play a crucial role in applied algebraic geometry and have applications in various fields, including cryptography, coding theory, and robotics