# Polynomials that lift #

Given semirings R and S with a morphism f : R →+* S, we define a subsemiring lifts of S[X] by the image of RingHom.of (map f). Then, we prove that a polynomial that lifts can always be lifted to a polynomial of the same degree and that a monic polynomial that lifts can be lifted to a monic polynomial (of the same degree).

## Main definition #

• lifts (f : R →+* S) : the subsemiring of polynomials that lift.

## Main results #

• lifts_and_degree_eq : A polynomial lifts if and only if it can be lifted to a polynomial of the same degree.
• lifts_and_degree_eq_and_monic : A monic polynomial lifts if and only if it can be lifted to a monic polynomial of the same degree.
• lifts_iff_alg : if R is commutative, a polynomial lifts if and only if it is in the image of mapAlg, where mapAlg : R[X] →ₐ[R] S[X] is the only R-algebra map that sends X to X.

## Implementation details #

In general R and S are semiring, so lifts is a semiring. In the case of rings, see lifts_iff_lifts_ring.

Since we do not assume R to be commutative, we cannot say in general that the set of polynomials that lift is a subalgebra. (By lift_iff this is true if R is commutative.)

def Polynomial.lifts {R : Type u} [] {S : Type v} [] (f : R →+* S) :

We define the subsemiring of polynomials that lifts as the image of RingHom.of (map f).

Equations
• = .rangeS
Instances For
theorem Polynomial.mem_lifts {R : Type u} [] {S : Type v} [] {f : R →+* S} (p : ) :
∃ (q : ), = p
theorem Polynomial.lifts_iff_set_range {R : Type u} [] {S : Type v} [] {f : R →+* S} (p : ) :
p
theorem Polynomial.lifts_iff_ringHom_rangeS {R : Type u} [] {S : Type v} [] {f : R →+* S} (p : ) :
p .rangeS
theorem Polynomial.lifts_iff_coeff_lifts {R : Type u} [] {S : Type v} [] {f : R →+* S} (p : ) :
∀ (n : ), p.coeff n
theorem Polynomial.C_mem_lifts {R : Type u} [] {S : Type v} [] (f : R →+* S) (r : R) :
Polynomial.C (f r)

If (r : R), then C (f r) lifts.

theorem Polynomial.C'_mem_lifts {R : Type u} [] {S : Type v} [] {f : R →+* S} {s : S} (h : s ) :
Polynomial.C s

If (s : S) is in the image of f, then C s lifts.

theorem Polynomial.X_mem_lifts {R : Type u} [] {S : Type v} [] (f : R →+* S) :
Polynomial.X

The polynomial X lifts.

theorem Polynomial.X_pow_mem_lifts {R : Type u} [] {S : Type v} [] (f : R →+* S) (n : ) :
Polynomial.X ^ n

The polynomial X ^ n lifts.

theorem Polynomial.base_mul_mem_lifts {R : Type u} [] {S : Type v} [] {f : R →+* S} {p : } (r : R) (hp : ) :
Polynomial.C (f r) * p

If p lifts and (r : R) then r * p lifts.

theorem Polynomial.monomial_mem_lifts {R : Type u} [] {S : Type v} [] {f : R →+* S} {s : S} (n : ) (h : s ) :

If (s : S) is in the image of f, then monomial n s lifts.

theorem Polynomial.erase_mem_lifts {R : Type u} [] {S : Type v} [] {f : R →+* S} {p : } (n : ) (h : ) :

If p lifts then p.erase n lifts.

theorem Polynomial.monomial_mem_lifts_and_degree_eq {R : Type u} [] {S : Type v} [] {f : R →+* S} {s : S} {n : } (hl : ) :
∃ (q : ), = s q.degree = ( s).degree
theorem Polynomial.mem_lifts_and_degree_eq {R : Type u} [] {S : Type v} [] {f : R →+* S} {p : } (hlifts : ) :
∃ (q : ), = p q.degree = p.degree

A polynomial lifts if and only if it can be lifted to a polynomial of the same degree.

theorem Polynomial.lifts_and_degree_eq_and_monic {R : Type u} [] {S : Type v} [] {f : R →+* S} [] {p : } (hlifts : ) (hp : p.Monic) :
∃ (q : ), = p q.degree = p.degree q.Monic

A monic polynomial lifts if and only if it can be lifted to a monic polynomial of the same degree.

theorem Polynomial.lifts_and_natDegree_eq_and_monic {R : Type u} [] {S : Type v} [] {f : R →+* S} {p : } (hlifts : ) (hp : p.Monic) :
∃ (q : ), = p q.natDegree = p.natDegree q.Monic
def Polynomial.liftsRing {R : Type u} [Ring R] {S : Type v} [Ring S] (f : R →+* S) :

The subring of polynomials that lift.

Equations
• = .range
Instances For
theorem Polynomial.lifts_iff_liftsRing {R : Type u} [Ring R] {S : Type v} [Ring S] (f : R →+* S) (p : ) :

If R and S are rings, p is in the subring of polynomials that lift if and only if it is in the subsemiring of polynomials that lift.

def Polynomial.mapAlg (R : Type u) [] (S : Type v) [] [Algebra R S] :

The map R[X] → S[X] as an algebra homomorphism.

Equations
Instances For
theorem Polynomial.mapAlg_eq_map {R : Type u} [] {S : Type v} [] [Algebra R S] (p : ) :
() p = Polynomial.map () p

mapAlg is the morphism induced by R → S.

theorem Polynomial.mem_lifts_iff_mem_alg (R : Type u) [] {S : Type v} [] [Algebra R S] (p : ) :
p p ().range

A polynomial p lifts if and only if it is in the image of mapAlg.

theorem Polynomial.smul_mem_lifts {R : Type u} [] {S : Type v} [] [Algebra R S] {p : } (r : R) (hp : p ) :
r p

If p lifts and (r : R) then r • p lifts.