Let $R$ be a non-commutative ring. Then we can define the semigroup ring of $R$ over the free non-commutative semigroup $F^{\rightarrow}(A)$. Then this semigroup ring $RF^{\rightarrow}(A)$ consists of ordered words over the alphabet $A$ with coefficients in $R$. This mirrors the construction of the polynomial ring $R[x_1,x_2,...]$ in commutative algebra as a free commutative semigroup ring.
Elements of the free non-commutative semigroup ring $RF^{\rightarrow}(x,y,z)$ consist of ordered words with coefficients in $R$. For example, we might get a polynomial like $5xyx + 6xz + 7zx$. A term like $xyx$ is not the same as $x^2y$ and $xz$ and $zx$ are not the same as one another. This non-commutativity would naturally add a great deal of complexity to compuations over non-commutative rings.
The issue with this free semigroup ring, is that it is not truly the ring of polynomial functions over $R$. Consider $c_1,c_2,...$ to be constants and $x,y,z,...$ to be indeterminates. Then a coefficient may not commute with its indeterminate, and $c_1 x \not= xc_1$. So for polynomial functions over a non-commutative ring you ultimately get potentially nasty terms like $c_1 x c_2 c_3 y c_4 x c_5 z c_6$.
A term of a polynomial function in a non-commutative ring is an alternating sequence of coefficients and indeterminates, where the coefficients can be identities. So for example $c_1xyc_2yxc_3$ is a term except the coefficients between $x$ and $y$ and then between $y$ and $x$ are simply identities, but there is always the possibility of having coefficients between indeterminates.
The terms of the free semigroup ring are already complicated enough, but terms in the ring of polynomial functions with scattered coefficients are even more messy. This is a serious impediment to doing noncommutative algebraic geometry, so what is the big idea? It seems that noncommutative geometry has more to do with functional analysis and operator theory then polynomials.
Noncommutative spaces can instead be studied using concepts of functional analysis and operator theory like C*-algebras. In particular, the Gelfand representation related C* algebras to locally compact Hausdorff spaces. The study of noncommutative geometry based upon techniques from functional analysis is an interesting direction, albiet one that is very different from you might expect.
Showing posts with label polynomials. Show all posts
Showing posts with label polynomials. Show all posts
Friday, September 24, 2021
Wednesday, September 22, 2021
Rings of multivariable Laurent polynomials
The commutative group ring of the free commutative group $F^{\circ}(X)$ over a field $k$ is a ring extension of the commutative semigroup ring of the free commutative monoid $F(X)$ over $k$. Although, $F^{\circ}(X)$ is a natural semigroup extension of $F(X)$, rings of multivariable Laurent polynomials don't have the same level of importance as polynomial rings in commutative algebra.
A distinguishing property of ordinary polynomials is that they can be cast into functions, so that over affine space $\mathbb{A}^n$ they are functions $f: \mathbb{A}^n \to k$ and this works for any commutative ring. On the other hand, for Laurent polynomials to be cast into functions $f : \mathbb{A}^n \to k$ we need to have some notion of division to deal with negative degree exponents, so we need to work over a field.
In those cases when we take the commutative group ring of a commutative ring $R$ which is not a field, with respect to $F^{\circ}(X)$ then what we get is certainly still a commutative ring, it is just not a commutative ring of functions. Consider the commutative group ring $\mathbb{Z}(\mathbb{Z}^n,+)$ consisting of polynomials with integer exponents and integer coefficients. Then this is a valid commutative ring, but its elements are not functions because $\mathbb{Z}$ is not a field.
So in some sense we ought to have our free commutative group ring be over a field $k$. Then we get terms like $5\frac{x}{yz} + \frac{6y}{xz}$ consisting of fractional monomials, but then given any term like this we can add to get $\frac{5x^2 + 6y^2}{xyz}$ which is always a polynomial with a monomial in the denominator. So we see that multivariable Laurent polynomials are simply rational functions with monomials in the denominator, so we have an embedding. \[ kF^{\circ}(x,y,z) \subseteq k(x,y,z) \] The ring of Laurent polynomials is simply the localisation of the polynomial ring by the multiplicative set of monomials, and so they are merely a special case of rational functions. Now it is clear why we don't see Laurent polynomials so much in commutative alebra. They are simply part of the far more important and general concept of fields of rational functions $k(x,y,z)$.
Just as we don't tend to restricted localisations of the integers like the dyadic rationals that much, we won't see rings of multivariable Laurent polynomials showing up as much. In general, we always want to deal with the largest localisation of a domain, which is its field of fractions.
So although the group completion of the free commutative semigroup $F(X)$ is an important and natural concept in commutative semigroup theory, it doesn't have the same role and level of importance in commutative algebra, as determined by commutative semigroup rings. This demonstrates that not everything in commutative algebra is a consequence of commutative semigroup rings, but the use of semigroup rings is still an infinitely powerful technique in the construction of commutative rings.
A distinguishing property of ordinary polynomials is that they can be cast into functions, so that over affine space $\mathbb{A}^n$ they are functions $f: \mathbb{A}^n \to k$ and this works for any commutative ring. On the other hand, for Laurent polynomials to be cast into functions $f : \mathbb{A}^n \to k$ we need to have some notion of division to deal with negative degree exponents, so we need to work over a field.
In those cases when we take the commutative group ring of a commutative ring $R$ which is not a field, with respect to $F^{\circ}(X)$ then what we get is certainly still a commutative ring, it is just not a commutative ring of functions. Consider the commutative group ring $\mathbb{Z}(\mathbb{Z}^n,+)$ consisting of polynomials with integer exponents and integer coefficients. Then this is a valid commutative ring, but its elements are not functions because $\mathbb{Z}$ is not a field.
So in some sense we ought to have our free commutative group ring be over a field $k$. Then we get terms like $5\frac{x}{yz} + \frac{6y}{xz}$ consisting of fractional monomials, but then given any term like this we can add to get $\frac{5x^2 + 6y^2}{xyz}$ which is always a polynomial with a monomial in the denominator. So we see that multivariable Laurent polynomials are simply rational functions with monomials in the denominator, so we have an embedding. \[ kF^{\circ}(x,y,z) \subseteq k(x,y,z) \] The ring of Laurent polynomials is simply the localisation of the polynomial ring by the multiplicative set of monomials, and so they are merely a special case of rational functions. Now it is clear why we don't see Laurent polynomials so much in commutative alebra. They are simply part of the far more important and general concept of fields of rational functions $k(x,y,z)$.
Just as we don't tend to restricted localisations of the integers like the dyadic rationals that much, we won't see rings of multivariable Laurent polynomials showing up as much. In general, we always want to deal with the largest localisation of a domain, which is its field of fractions.
So although the group completion of the free commutative semigroup $F(X)$ is an important and natural concept in commutative semigroup theory, it doesn't have the same role and level of importance in commutative algebra, as determined by commutative semigroup rings. This demonstrates that not everything in commutative algebra is a consequence of commutative semigroup rings, but the use of semigroup rings is still an infinitely powerful technique in the construction of commutative rings.
Wednesday, December 16, 2020
Ontology of polynomials
When considering polynomials over a commutative ring, it is often useful to limit yourself to considering special cases. Even though the most interesting polynomials are multivariable, we tend to limit ourselves to univariate polynomials or even more then that to monic polynomials such as in the definition of integral extensions. The whole field of linear algebra basically built around the simple idea of a linear polynomial. Clearly, a classification system for polynomials is necessary. Here are some classes to start with:
Polynomials are relatively simple data structures, which make them a solid part of any computer algebra system. They can simply be represented by a collection of individual monomial terms, which only need to contain a coefficient and some variables. With such a simple representation as a sum of terms, each of these different classes of polynomials can easily be turned into computable predicates on polynomials. This is then a hierarchy of computable predicates, which is generally the nicest kind of ontology.
Homogeneous polynomials: it is trivial to check if a given polynomial has only terms of the same degree, degree in this case means the sum of all exponents of variables in each term. $x^2 + y^2 + xy$ for example is a homogeneous polynomial and a binary quadratic form while $ax^3 + bx^2y + cxy^2 + dy^3$ is a binary cubic form. Homogeneous polynomials are fundamental in algebraic geometry because roots are invariant under scaling, which produces a natural link to projective geometry. Applying a little creativity, we could define anti-homogeneous polynomials to be ones with different degrees for each term.
Max power one polynomials: these are polynomials in which the exponent of each variable in each term is never greater then one. For example $xy + yz + xz$ is a max power one quadratic form but $x^2 + 2x + 1$ is not because of the exponent of two in one of the variables. Although not as familiar, these might appear from semirings in which multiplication is idempotent.
Diagonal polynomials: the dual of a polynomial in which each variable is different, is one in which each variable is the same. We can call these diagonal polynomials and diagonal forms are special cases that are also homogeneous. For example, $x^3 + y^3 + z^3$ is a diagonal form.
Additional classes: this is a very limited upper ontology, which is applicable to general rings. We could classify polynomials themselves based upon the ring that they emerge from, so for example we could consider real polynomials, complex polynomials, etc but these wouldn't fit into an upper ontology. As these classes are based upon sum representation, factorisation based considerations like separable and irreducible polynomials are not included. In algebraically closed fields, irreducible polynomials are simply linear univariate ones, so over some rings it is not necessary to consider irreducible polynomials as a separate class, so they must dealt with separately.
Homogeneous polynomials: it is trivial to check if a given polynomial has only terms of the same degree, degree in this case means the sum of all exponents of variables in each term. $x^2 + y^2 + xy$ for example is a homogeneous polynomial and a binary quadratic form while $ax^3 + bx^2y + cxy^2 + dy^3$ is a binary cubic form. Homogeneous polynomials are fundamental in algebraic geometry because roots are invariant under scaling, which produces a natural link to projective geometry. Applying a little creativity, we could define anti-homogeneous polynomials to be ones with different degrees for each term.
Max power one polynomials: these are polynomials in which the exponent of each variable in each term is never greater then one. For example $xy + yz + xz$ is a max power one quadratic form but $x^2 + 2x + 1$ is not because of the exponent of two in one of the variables. Although not as familiar, these might appear from semirings in which multiplication is idempotent.
Diagonal polynomials: the dual of a polynomial in which each variable is different, is one in which each variable is the same. We can call these diagonal polynomials and diagonal forms are special cases that are also homogeneous. For example, $x^3 + y^3 + z^3$ is a diagonal form.
Additional classes: this is a very limited upper ontology, which is applicable to general rings. We could classify polynomials themselves based upon the ring that they emerge from, so for example we could consider real polynomials, complex polynomials, etc but these wouldn't fit into an upper ontology. As these classes are based upon sum representation, factorisation based considerations like separable and irreducible polynomials are not included. In algebraically closed fields, irreducible polynomials are simply linear univariate ones, so over some rings it is not necessary to consider irreducible polynomials as a separate class, so they must dealt with separately.
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