nLab degree of a polynomial

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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 pp in the form of the scalar coefficients of pp.

Polynomials in one indeterminate

Given a commutative ring RR, the commutative ring of polynomials R[x]R[x] in one indeterminate xx is the initial commutative ring R[x]R[x] with an element xR[x]x \in R[x] and a ring homomorphism h:RR[x]h:R \to R[x].

Using derivatives

The formal derivative d()dx:R[x]R[x]\frac{d (-)}{d x}:R[x] \to R[x] is a function which is inductively defined on R[x]R[x] by

  • Constants go to zero: dh(r)dx=0\frac{d h(r)}{d x} = 0 for all rRr \in R
  • Indeterminate goes to one: dxdx=h(1)\frac{d x}{d x} = h(1)
  • Addition preservation: d(p+q)dx=dpdx+dqdx\frac{d (p + q)}{d x} = \frac{d p}{d x} + \frac{d q}{d x} for all pR[x]p \in R[x] and qR[x]q \in R[x].
  • Leibniz rule: d(pq)dx=dqdxq+pdqdx\frac{d (p \cdot q)}{d x} = \frac{d q}{d x} \cdot q + p \cdot \frac{d q}{d x} for all pR[x]p \in R[x] and qR[x]q \in R[x].

By the universal property of R[x]R[x], these constructors are enough to define the derivative on all polynomials pR[x]p \in R[x].

Let d ()()dx ():(R[X]R[X])\frac{d^{(-)} (-)}{d x^{(-)}}:\mathbb{N} \to (R[X] \to R[X]) be the function which takes a natural number nn \in \mathbb{N} to the nn-th derivative in the function set R[X]R[X]R[X] \to R[X]. This is inductively defined on the natural numbers by

  • d 0()dx 0=id R[X]\frac{d^{0} (-)}{d x^{0}} = \mathrm{id}_{R[X]}
  • d s(n)()dx s(n)=d()dxd n()dx n\frac{d^{s(n)} (-)}{d x^{s(n)}} = \frac{d (-)}{d x} \circ \frac{d^{n} (-)}{d x^{n}} for all nn \in \mathbb{N}

If RR has characteristic zero, given a non-zero polynomial pR[x]p \in R[x], the degree of pp is the maximum natural number nn where the nn-th iteration of the formal derivative of pp is non-zero:

deg(p)max n,d npdx n0(n)\mathrm{deg}(p) \coloneqq \max_{n \in \mathbb{N}, \frac{d^{n} p}{d x^{n}} \neq 0}(n)

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 RR with characteristic zero. However, if RR has positive characteristic, the formal derivative is problematic for the degree, because given characteristic nn, the derivative of the nn-th power of the indeterminate xx is zero. Instead, we have to use something more general, the left shift.

Using left shifts

For every polynomial pR[x]p \in R[x], pp could be represented as the sum of a constant polynomial c pc_p and the product of the indeterminate xx and a polynomial s L(p)s_L(p) called the left shift of pp: p=c p+xs L(p)p = c_p + x \cdot s_L(p). Given any two constant polynomials p=c pp = c_p and q=c qq = c_q, by definition, c pq=c pc qc_{p \cdot q} = c_p \cdot c_q. Now, given any two general polynomials p=c p+xs L(p)p = c_p + x \cdot s_L(p) and q=c q+xs L(q)q = c_q + x \cdot s_L(q), one then has the following, through the axioms of a commutative ring:

p=xs L(p)+c p,q=xs L(q)+c qp = x \cdot s_L(p) + c_p, q = x \cdot s_L(q) + c_q
pq=xs L(pq)+c pq=xs L(pq)+c pc qp \cdot q = x \cdot s_L(p \cdot q) + c_{p \cdot q} = x s_L(p \cdot q) + c_{p} \cdot c_{q}
pq=(xs L(p)+c p)(xs L(q)+c q)=x 2s L(p)s L(q)+xs L(p)c q+xs L(q)c p+c pc qp \cdot q = (x \cdot s_L(p) + c_p) \cdot (x \cdot s_L(q) + c_q) = x^2 \cdot s_L(p) \cdot s_L(q) + x \cdot s_L(p) \cdot c_q + x \cdot s_L(q) \cdot c_p + c_p \cdot c_q
xs L(pq)=x 2s L(p)s L(q)+xs L(p)c q+xs L(q)c px \cdot s_L(p \cdot q) = x^2 \cdot s_L(p) \cdot s_L(q) + x \cdot s_L(p) \cdot c_q + x \cdot s_L(q) \cdot c_p
s L(pq)=xs L(p)s L(q)+s L(p)c q+s L(q)c ps_L(p \cdot q) = x \cdot s_L(p) \cdot s_L(q) + s_L(p) \cdot c_q + s_L(q) \cdot c_p
pxs L(p)=c p,qxs L(q)=c qp - x \cdot s_L(p) = c_p, q - x \cdot s_L(q) = c_q
s L(pq)=xs L(p)s L(q)+s L(p)(qxs L(q))+s L(q)(pxs L(p))=s L(p)q+s L(q)pxs L(p)s L(q)s_L(p \cdot q) = x \cdot s_L(p) \cdot s_L(q) + s_L(p) \cdot (q - x \cdot s_L(q)) + s_L(q) \cdot (p - x \cdot s_L(p)) = s_L(p) \cdot q + s_L(q) \cdot p - x \cdot s_L(p) \cdot s_L(q)

Thus, we have the formula for the left shift of the product pqp \cdot q for any two polynomials pR[x]p \in R[x] and qR[x]q \in R[x]:

s L(pq)=s L(p)q+ps L(q)xs L(p)s L(q)s_L(p \cdot q) = s_L(p) \cdot q + p \cdot s_L(q) - x \cdot s_L(p) \cdot s_L(q)

The only difference from the Leibniz rule for the formal derivative is the extra term xs L(p)s L(q)-x \cdot s_L(p) \cdot s_L(q) at the end of the formula.

This allows us to inductively define on R[x]R[x] the formal left shift operator s L:R[x]R[x]s_L:R[x] \to R[x] in the same manner as the formal derivative, by

  • Constants go to zero: s L(h(r))=0s_L(h(r)) = 0 for all rRr \in R
  • Indeterminate goes to one: s L(x)=h(1)s_L(x) = h(1)
  • Addition preservation: s L(p+q)=s L(p)+s L(q)s_L(p + q) = s_L(p) + s_L(q) for all p:R[x]p:R[x] and q:R[x]q:R[x].
  • Left shift product rule: s L(pq)=s L(p)q+ps L(q)xs L(p)s L(q)s_L(p \cdot q) = s_L(p) \cdot q + p \cdot s_L(q) - x \cdot s_L(p) \cdot s_L(q) for all pR[x]p \in R[x] and qR[x]q \in R[x].

By the universal property of R[x]R[x], these constructors are enough to define the left shift on all polynomials pR[x]p \in R[x].

Let s L n:(R[X]R[X])s_L^n:\mathbb{N} \to (R[X] \to R[X]) be the function which takes a natural number nn \in \mathbb{N} to the nn-th left shift in the function set R[X]R[X]R[X] \to R[X]. This is inductively defined on the natural numbers by the following constructors

  • s L 0=id R[X]s_L^0 = \mathrm{id}_{R[X]}
  • s L s(n)=s Ls L ns_L^{s(n)} = s_L \circ s_L^n for all nn \in \mathbb{N}

In this defintion, RR is not required to have characteristic zero. Given a non-zero polynomial pR[x]p \in R[x], the degree of pp is the maximum natural number nn where the nn-th iteration of the formal left shift operator of pp is non-zero:

deg(p)max n,s L n(p)0(n)\mathrm{deg}(p) \coloneqq \max_{n \in \mathbb{N}, s_L^n(p) \neq 0}(n)

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 Fin(n)\mathrm{Fin}(n) as the set of all natural numbers less than nn:

Fin(n){i|i<n}\mathrm{Fin}(n) \coloneqq \{i \in \mathbb{N} \vert i \lt n\}

Given a natural number less than nn, i:Fin(n)i:\mathrm{Fin}(n), let Fin(n){i}\mathrm{Fin}(n) \setminus \{i\} be the set of natural numbers less than nn which are not equal to ii.

Given a commutative ring RR with characteristic zero and a natural number nn, the commutative ring of polynomials R[X]R[X] is the initial commutative ring R[X]R[X] with a function X:Fin(n)R[X]X:\mathrm{Fin}(n) \to R[X] and a ring homomorphism h:RR[X]h:R \to R[X]. We write X iX_i for X(i)X(i) throughout. Given a natural number less than nn, iFin(n)i \in \mathrm{Fin}(n), let R[X{X i}]R[X \setminus \{X_i\}] be the polynomial subring of R[X]R[X] whose indeterminates do not include X iX_i, with monomorphism m:R[X{X i}]R[X]m:R[X \setminus \{X_i\}] \hookrightarrow R[X].

Let Fin(n) *\mathrm{Fin}(n)^* be the free monoid on the set of natural numbers less than nn. Every free monoid Fin(n) *\mathrm{Fin}(n)^* has a length function len:Fin(n) *\mathrm{len}:\mathrm{Fin}(n)^* \to \mathbb{N} which returns the number of elements in a list aFin(n) *a \in \mathrm{Fin}(n)^*, inductively defined by the following constructors

  • Monoidal unit preservation: len(ϵ)=0\mathrm{len}(\epsilon) = 0
  • Monoidal product preservation: len(ab)=len(a)+len(b)\mathrm{len}(a b) = \mathrm{len}(a) + \mathrm{len}(b) for all aFin(n) *a \in \mathrm{Fin}(n)^* and bFin(n) *b \in \mathrm{Fin}(n)^*
  • Generators have length 1: len(i)=1\mathrm{len}(i) = 1 for all iFin(n)i \in \mathrm{Fin}(n)

With partial derivatives

The formal partial derivative ()X ():Fin(n)×R[X]R[X]\frac{\partial(-)}{\partial X_{(-)}}:\mathrm{Fin}(n) \times R[X] \to R[X] is inductively defined by

  • Constants relative to the indeterminant go to zero: h(r)X i=0\frac{\partial h(r)}{\partial X_i} = 0 for all iFin(n)i \in \mathrm{Fin}(n) and rR[X{X i}]r \in R[X \setminus \{X_i\}]
  • Indeterminates go to one: X iX i=1\frac{\partial X_i}{\partial X_i} = 1 for all iFin(n)i \in \mathrm{Fin}(n)
  • Addition preservation: (P+Q)X i=PX i+QX i\frac{\partial (P + Q)}{\partial X_i} = \frac{\partial P}{\partial X_i} + \frac{\partial Q}{\partial X_i} for all iFin(n)i \in \mathrm{Fin}(n), PR[X]P \in R[X] and QR[X]Q \in R[X].
  • Leibniz rule: (PQ)X i=PX iQ+PQX i\frac{\partial (P \cdot Q)}{\partial X_i} = \frac{\partial P}{\partial X_i} \cdot Q + P \cdot \frac{\partial Q}{\partial X_i} for all iFin(n)i \in \mathrm{Fin}(n), PR[X]P \in R[X] and QR[X]Q \in R[X].

By the universal property of R[X]R[X], these constructors are enough to define the partial derivatives on all polynomials PR[X]P \in R[X].

Let d:Fin(n) *(R[X]R[X])d:\mathrm{Fin}(n)^* \to (R[X] \to R[X]) be the monoid homomorphism which takes a list aFin(n) *a \in \mathrm{Fin}(n)^* of natural numbers less than nn, to the composition of formal partial derivatives

d(a) len(a)()X a 0X a 1X a 2()X a 0()X a 1()X a 2d(a) \coloneqq \frac{\partial^{\mathrm{len}(a)} (-)}{\partial X_{a_0} \partial X_{a_1} \partial X_{a_2} \ldots} \coloneqq \frac{\partial (-)}{\partial X_{a_0}} \circ \frac{\partial (-)}{\partial X_{a_1}} \circ \frac{\partial (-)}{\partial X_{a_2}} \circ \ldots

in the function set R[X]R[X]R[X] \to R[X]. This is inductively defined by the following constructors

  • Monoidal unit preservation: d(ϵ)=id R[X]d(\epsilon) = \mathrm{id}_{R[X]}
  • Monoidal product preservation: d(ab)=d(a)d(b)d(a b) = d(a) \circ d(b) for all aFin(n) *a \in \mathrm{Fin}(n)^* and bFin(n) *b \in \mathrm{Fin}(n)^*
  • Generators to partial derivatives: d(i)=()X id(i) = \frac{\partial (-)}{\partial X_i} for all iFin(n)i \in \mathrm{Fin}(n)

If RR has characteristic zero, given a non-zero polynomial PR[x]P \in R[x], the degree of PP is the maximum length of all lists aFin(n) *a \in \mathrm{Fin}(n)^* of natural numbers less than nn such that the evaluation of d(a)d(a) at PP is non-zero

deg(P)max aFin(n) *,d(a)(P)0(len(a))\mathrm{deg}(P) \coloneqq \max_{a \in \mathrm{Fin}(n)^*, d(a)(P) \neq 0}(\mathrm{len}(a))

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 RR with characteristic zero. However, if RR has positive characteristic, the formal derivative is problematic for the degree, because given characteristic pp, the derivative of the pp-th power of any indeterminate X iX_i for natural number less than nn, iFin(n)i \in \mathrm{Fin}(n), is zero. Instead, we have to use something more general, partial left shifts.

With partial left shifts

The formal partial left shift operator s L:Fin(n)×R[X]R[X]s_{\partial L}:\mathrm{Fin}(n) \times R[X] \to R[X] is inductively defined by

  • Constants relative to the indeterminant go to zero: s L(i,r)=0s_{\partial L}(i, r) = 0 for all iFin(n)i \in \mathrm{Fin}(n) and rR[X{X i}]r \in R[X \setminus \{X_i\}]
  • Indeterminates go to one: s L(i,X i)=1s_{\partial L}(i, X_i) = 1 for all iFin(n)i \in \mathrm{Fin}(n)
  • Addition preservation: s L(i,P+Q)=s L(i,P)+s L(i,Q)s_{\partial L}(i, P + Q) = s_{\partial L}(i, P) + s_{\partial L}(i, Q) for all iFin(n)i \in \mathrm{Fin}(n), PR[X]P \in R[X] and QR[X]Q \in R[X].
  • Left shift product rule: s L(i,PQ)=s L(i,P)Q+Ps L(i,Q)X is L(i,P)s L(i,Q)s_{\partial L}(i, P \cdot Q) = s_{\partial L}(i, P) \cdot Q + P \cdot s_{\partial L}(i, Q) - X_i \cdot s_{\partial L}(i, P) \cdot s_{\partial L}(i, Q) for all iFin(n)i \in \mathrm{Fin}(n), pR[x]p \in R[x] and qR[x]q \in R[x].

By the universal property of R[X]R[X], these constructors are enough to define the partial left shifts on all polynomials PR[X]P \in R[X].

Let S:Fin(n) *(R[X]R[X])S:\mathrm{Fin}(n)^* \to (R[X] \to R[X]) be the monoid homomorphism which takes a list aFin(n) *a\in \mathrm{Fin}(n)^* of natural numbers less than nn, to the composition of formal partial left shifts

S(a)s L(a 0,)s L(a 1,)s L(a 2,)S(a) \coloneqq s_{\partial L}(a_0, -) \circ s_{\partial L}(a_1, -) \circ s_{\partial L}(a_2, -) \circ \ldots

in the function set R[X]R[X]R[X] \to R[X]. This is inductively defined by the following constructors

  • Monoidal unit preservation: S(ϵ)=id R[X]S(\epsilon) = \mathrm{id}_{R[X]}
  • Monoidal product preservation: S(ab)=S(a)S(b)S(a b) = S(a) \circ S(b) for all aFin(n) *a \in \mathrm{Fin}(n)^* and bFin(n) *b \in \mathrm{Fin}(n)^*
  • Generators to partial left shifts S(i)=s L(i,)S(i) = s_{\partial L}(i, -) for all iFin(n)i \in \mathrm{Fin}(n)

Given a non-zero polynomial PR[x]P \in R[x], the degree of PP is the maximum length of all lists aFin(n) *a \in \mathrm{Fin}(n)^* of natural numbers less than nn such that the evaluation of S(a)S(a) at PP is non-zero

deg(P)max aFin(n) *,S(a)(P)0(len(a))\mathrm{deg}(P) \coloneqq \max_{a \in \mathrm{Fin}(n)^*, S(a)(P) \neq 0}(\mathrm{len}(a))

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 pR[x]p \in R[x] in one indeterminate, it is said to be an homogeneous polynomial of degree nn if dpdx.x=n.p\frac{d p}{d x}.x = n.p (see the Euler identity).

Given a polynomial pR[X]p \in R[X] in a finite number of indeterminates, it is said to be an homogeneous polynomial of degree nn if iFin(n)pX i.X i=n.p\underset{i \in \mathrm{Fin}(n)}{\sum} \frac{\partial p}{\partial X_i}.X_i = n.p.

One could also define a polynomial pR[X]p \in R[X] to be homogeneous of degree nn with respect to iFin(n)i \in \mathrm{Fin}(n) if pX i.X i=n.p\frac{\partial p}{\partial X_i}.X_i = n.p. It is equivalent to the fact that pp can be written under the form p=X i n.qp=X_i^{n}.q where qR[X]q \in R[X] is such that pX i=0\frac{\partial p}{\partial X_i} = 0.

 In constructive mathematics

In constructive mathematics, where excluded middle does not hold, the above definition is correct only if the ring RR has decidable equality. In more general circumstances, one has to assume that the ring RR and thus the polynomial ring R[X]R[X] is an inequality space with a tight apartness relation #\#, and replace all instances of “non-zero” p0p \neq 0 with instances of “apart from zero” p#0p \# 0. (That every set RR 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

Last revised on November 24, 2022 at 13:48:59. See the history of this page for a list of all contributions to it.