3.1 Multivariate polynomials and polynomial rings

Multivariate polynomials are like mathematical superheroes, juggling multiple variables at once. They're the building blocks of polynomial rings, letting us create complex mathematical structures. These polynomials and rings are the dynamic duo of algebraic geometry.

Polynomial rings are where the magic happens. They're the playground for multivariate polynomials, with rules for addition and multiplication. Understanding these rings helps us solve tricky equations and explore geometric shapes in higher dimensions.

Multivariate Polynomials

Definition and Properties

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• Multivariate polynomials are polynomials in two or more variables, with coefficients typically drawn from a field or ring (e.g., real numbers, complex numbers, or integers)
• The degree of a multivariate polynomial is the highest total degree of any term
  • The total degree of a term is the sum of the exponents of all variables in that term (e.g., in the term $3x^2y^3$, the total degree is $2 + 3 = 5$)
• Multivariate polynomials can be added, subtracted, and multiplied using similar rules as single-variable polynomials, with additional consideration for multiple variables
• Division of multivariate polynomials is more complex than single-variable polynomials due to the notion of degree and the choice of monomial ordering affecting the outcome
• Evaluating multivariate polynomials involves substituting values for the variables, with the resulting value depending on the specific values chosen for each variable (e.g., evaluating $f(x, y) = x^2 + 2xy + y^2$ at $(1, 2)$ gives $f(1, 2) = 1^2 + 2(1)(2) + 2^2 = 9$)

Operations and Evaluation

• Addition and subtraction of multivariate polynomials involve combining like terms, considering the coefficients and exponents of all variables (e.g., $(2x^2y + 3xy^2) + (4x^2y - xy^2) = 6x^2y + 2xy^2$)
• Multiplication of multivariate polynomials follows the distributive law and involves multiplying each term of one polynomial by each term of the other polynomial (e.g., $(x + y)(x - y) = x^2 - y^2$)
• Division of multivariate polynomials requires choosing a monomial ordering, which affects the quotient and remainder (e.g., dividing $x^2y + xy^2 + y$ by $xy + 1$ using lexicographic order gives quotient $xy - y + 1$ and remainder $y^2$)
• Evaluating multivariate polynomials at specific points or sets of points is useful for understanding their behavior and solving systems of polynomial equations (e.g., finding the zeros of a polynomial system $f(x, y) = 0$ and $g(x, y) = 0$)

Polynomial Rings

Construction and Definition

• A polynomial ring is a ring formed by the set of all polynomials in one or more variables with coefficients from a given ring or field (e.g., the polynomial ring $\mathbb{R}[x, y]$ consists of all polynomials in variables $x$ and $y$ with real coefficients)
• The polynomial ring in $n$ variables over a ring $R$ is denoted as $R[x_1, x_2, \ldots, x_n]$, where $x_1, x_2, \ldots, x_n$ are the variables and $R$ is the coefficient ring
• Elements of a polynomial ring are multivariate polynomials, and the ring operations (addition and multiplication) are defined as the usual addition and multiplication of polynomials
• Polynomial rings are commutative rings, meaning the multiplication operation is commutative: $f(x_1, \ldots, x_n) \cdot g(x_1, \ldots, x_n) = g(x_1, \ldots, x_n) \cdot f(x_1, \ldots, x_n)$ for any two polynomials $f$ and $g$

Ideals and Quotient Rings

• An ideal in a polynomial ring is a subset closed under addition and multiplication by ring elements (e.g., the set of all polynomials with constant term 0 forms an ideal in $\mathbb{R}[x]$)
• Principal ideals in a polynomial ring are ideals generated by a single polynomial (e.g., the ideal $\langle x^2 + 1 \rangle$ in $\mathbb{R}[x]$ consists of all polynomials divisible by $x^2 + 1$)
• Quotient rings of polynomial rings by ideals are rings obtained by "dividing out" the ideal, with elements being equivalence classes of polynomials (e.g., $\mathbb{R}[x] / \langle x^2 + 1 \rangle$ is a quotient ring where polynomials that differ by a multiple of $x^2 + 1$ are considered equivalent)
• Ideals and quotient rings play a crucial role in studying the structure of polynomial rings and their relation to algebraic varieties

Structure of Polynomial Rings

Integral Domains and Principal Ideal Domains

• Polynomial rings are integral domains, meaning they have no zero divisors: if $f(x_1, \ldots, x_n) \cdot g(x_1, \ldots, x_n) = 0$, then either $f(x_1, \ldots, x_n) = 0$ or $g(x_1, \ldots, x_n) = 0$
• Polynomial rings over a field are principal ideal domains (PIDs), meaning every ideal in the ring is generated by a single element (e.g., $\mathbb{C}[x, y]$ is a PID, but $\mathbb{Z}[x, y]$ is not)
• The Hilbert Basis Theorem states that every ideal in a polynomial ring over a field is finitely generated, making polynomial rings Noetherian rings

Gröbner Bases and Dimension

• Gröbner bases are special sets of generators for ideals in polynomial rings that provide a way to solve systems of polynomial equations and perform computations in quotient rings (e.g., the Gröbner basis of the ideal $\langle x^2 - y, x - z^2 \rangle$ in $\mathbb{Q}[x, y, z]$ with lexicographic order is $\{x - z^2, y - z^4\}$)
• The dimension of a polynomial ring is the number of variables in the ring, affecting the geometric and algebraic properties of the ring and its ideals (e.g., $\mathbb{R}[x, y, z]$ has dimension 3, while $\mathbb{R}[x]$ has dimension 1)
• Dimension is related to the complexity of the ring and the geometric objects it represents, with higher-dimensional rings often exhibiting more intricate behavior

Polynomials vs Rings

Relationship between Multivariate Polynomials and Polynomial Rings

• Multivariate polynomials are the elements of polynomial rings, and their properties determine the structure and characteristics of the rings
• The choice of coefficient ring or field affects the properties of the multivariate polynomials and the resulting polynomial ring (e.g., $\mathbb{Z}[x, y]$ has different properties than $\mathbb{Q}[x, y]$ due to the difference in coefficient rings)
• Ideals generated by multivariate polynomials in a polynomial ring correspond to algebraic varieties in affine or projective space, linking the algebraic properties of the ring to geometric objects (e.g., the ideal $\langle x^2 + y^2 - 1 \rangle$ in $\mathbb{R}[x, y]$ corresponds to the unit circle)
• Quotient rings of polynomial rings by ideals generated by multivariate polynomials represent the rings of functions on the corresponding algebraic varieties, connecting the ring-theoretic and geometric perspectives

Applications in Algebraic Geometry

• The study of multivariate polynomials and polynomial rings is fundamental to algebraic geometry, providing the algebraic foundation for investigating geometric properties of varieties and schemes
• Algebraic geometry uses the correspondence between ideals in polynomial rings and algebraic varieties to study geometric objects using algebraic techniques (e.g., studying the solutions of polynomial equations, intersections of varieties, and singular points)
• Techniques from commutative algebra, such as localization and completion of rings, are used to analyze local properties of varieties and construct schemes as locally ringed spaces
• The interplay between multivariate polynomials, polynomial rings, and their geometric counterparts is a central theme in algebraic geometry, with applications in various fields such as coding theory, cryptography, and mathematical physics