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Definition
The following definition of the degree of a polynomial is formulated vis the formal derivative or left shift of a polynomial.
The idea behind this lies in the traditional conception of polynomials as sums of monomials, namely of products of non-zero scalars with powers of an “indeterminate” – which have an associated exponent because commutative rings are power-associative in multiplication.
Both the derivative and the left shift are functions on polynomials which reduce the exponent of the associated indeterminate by one and take polynomials, relative to the given indeterminate, to zero. Since both operations are also linear functions, they act this way on each monomial summand, which means that after repeatedly taking derivatives or left shifts of polynomials, the result will eventually become zero.
This allows us to define the degree of a polynomial using induction on the natural numbers and on lists of natural numbers, as well as composition of left shifts, without having to write out long formulae using indices and ellipses everywhere in the definition, and without having to include additional structure with each polynomial in the form of the scalar coefficients of .
Polynomials in one indeterminate
Given a commutative ring , the commutative ring of polynomials in one indeterminate is the initial commutative ring with an element and a ring homomorphism .
Using derivatives
The formal derivative is a function which is inductively defined on by
- Constants go to zero: for all
- Indeterminate goes to one:
- Addition preservation: for all and .
- Leibniz rule: for all and .
By the universal property of , these constructors are enough to define the derivative on all polynomials .
Let be the function which takes a natural number to the -th derivative in the function set . This is inductively defined on the natural numbers by
- for all
If has characteristic zero, given a non-zero polynomial , the degree of is the maximum natural number where the -th iteration of the formal derivative of is non-zero:
The degree of the zero polynomial is undefined, since the formal derivative of the zero polynomial is never non-zero.
The above definition involving formal derivatives works for commutative rings with characteristic zero. However, if has positive characteristic, the formal derivative is problematic for the degree, because given characteristic , the derivative of the -th power of the indeterminate is zero. Instead, we have to use something more general, the left shift.
Using left shifts
For every polynomial , could be represented as the sum of a constant polynomial and the product of the indeterminate and a polynomial called the left shift of : . Given any two constant polynomials and , by definition, . Now, given any two general polynomials and , one then has the following, through the axioms of a commutative ring:
Thus, we have the formula for the left shift of the product for any two polynomials and :
The only difference from the Leibniz rule for the formal derivative is the extra term at the end of the formula.
This allows us to inductively define on the formal left shift operator in the same manner as the formal derivative, by
- Constants go to zero: for all
- Indeterminate goes to one:
- Addition preservation: for all and .
- Left shift product rule: for all and .
By the universal property of , these constructors are enough to define the left shift on all polynomials .
Let be the function which takes a natural number to the -th left shift in the function set . This is inductively defined on the natural numbers by the following constructors
- for all
In this defintion, is not required to have characteristic zero. Given a non-zero polynomial , the degree of is the maximum natural number where the -th iteration of the formal left shift operator of is non-zero:
The degree of the zero polynomial is undefined, since the left shift of the zero polynomial is never non-zero.
Polynomials in a finite number of indeterminates
Let us define the standard finite set as the set of all natural numbers less than :
Given a natural number less than , , let be the set of natural numbers less than which are not equal to .
Given a commutative ring with characteristic zero and a natural number , the commutative ring of polynomials is the initial commutative ring with a function and a ring homomorphism . We write for throughout. Given a natural number less than , , let be the polynomial subring of whose indeterminates do not include , with monomorphism .
Let be the free monoid on the set of natural numbers less than . Every free monoid has a length function which returns the number of elements in a list , inductively defined by the following constructors
- Monoidal unit preservation:
- Monoidal product preservation: for all and
- Generators have length 1: for all
With partial derivatives
The formal partial derivative is inductively defined by
- Constants relative to the indeterminant go to zero: for all and
- Indeterminates go to one: for all
- Addition preservation: for all , and .
- Leibniz rule: for all , and .
By the universal property of , these constructors are enough to define the partial derivatives on all polynomials .
Let be the monoid homomorphism which takes a list of natural numbers less than , to the composition of formal partial derivatives
in the function set . This is inductively defined by the following constructors
- Monoidal unit preservation:
- Monoidal product preservation: for all and
- Generators to partial derivatives: for all
If has characteristic zero, given a non-zero polynomial , the degree of is the maximum length of all lists of natural numbers less than such that the evaluation of at is non-zero
The degree of the zero polynomial is undefined, since any composition of partial derivatives evaluated at the zero polynomial is never non-zero.
The above definition involving formal partial derivatives works for commutative rings with characteristic zero. However, if has positive characteristic, the formal derivative is problematic for the degree, because given characteristic , the derivative of the -th power of any indeterminate for natural number less than , , is zero. Instead, we have to use something more general, partial left shifts.
With partial left shifts
The formal partial left shift operator is inductively defined by
- Constants relative to the indeterminant go to zero: for all and
- Indeterminates go to one: for all
- Addition preservation: for all , and .
- Left shift product rule: for all , and .
By the universal property of , these constructors are enough to define the partial left shifts on all polynomials .
Let be the monoid homomorphism which takes a list of natural numbers less than , to the composition of formal partial left shifts
in the function set . This is inductively defined by the following constructors
- Monoidal unit preservation:
- Monoidal product preservation: for all and
- Generators to partial left shifts for all
Given a non-zero polynomial , the degree of is the maximum length of all lists of natural numbers less than such that the evaluation of at is non-zero
The degree of the zero polynomial is undefined, since any composition of partial derivatives evaluated at the zero polynomial is never non-zero.
Homogeneous polynomials
…
Given a polynomial in one indeterminate, it is said to be an homogeneous polynomial of degree if (see the Euler identity).
Given a polynomial in a finite number of indeterminates, it is said to be an homogeneous polynomial of degree if .
One could also define a polynomial to be homogeneous of degree with respect to if . It is equivalent to the fact that can be written under the form where is such that .
In constructive mathematics
In constructive mathematics, where excluded middle does not hold, the above definition is correct only if the ring has decidable equality. In more general circumstances, one has to assume that the ring and thus the polynomial ring is an inequality space with a tight apartness relation , and replace all instances of “non-zero” with instances of “apart from zero” . (That every set is an inequality space in classical mathematics follows from the stability of decidable equality.) For example, the degree function on the Dedekind real numbers is only defined on polynomials with at least one coefficient whose absolute value is greater than zero.
See also