`Compfiles/Usa2019P6.lean`

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12 declarations, 0 contain sorry, 0 declared as axioms

USA Mathematical Olympiad 2019, Problem 6

Find all polynomials P with real coefficients such that

P(x)/yz + P(y)/zx + P(z)/xy = P(x − y) + P(y − z) + P(z − x)

for all nonzero real numbers x, y, z obeying 2xyz = x + y + z.

Imports (5)

Imported by 1 file

Compfiles.lean

determine Usa2019P6.SolutionSet line 33 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Set, Polynomial, Real
Proof uses: Set.ofPred, Exists, Eq, HMul.hMul, RingHom, Polynomial.C, HAdd.hAdd, HPow.hPow, Nat, Polynomial.X, OfNat.ofNat
18 instances, coercions and auxiliary constants left out.
def Usa2019P6.Qfun The polynomial equation, multiplied through by xyz line 40 source
Source documentation
The polynomial equation, multiplied through by xyz. This is the form of the hypothesis that we use below: it says that Q(x, y, z) vanishes, where Q(x, y, z) = xP(x) + yP(y) + zP(z) − xyz (P(x−y) + P(y−z) + P(z−x)).

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Field, Polynomial
Proof uses: HSub.hSub, HAdd.hAdd, HMul.hMul, Polynomial.eval
18 instances, coercions and auxiliary constants left out.
```
def Qfun {K : Type*} [Field K] (P : Polynomial K) (x y z : K) : K :=
…
```
Mentions: P
def Usa2019P6.PhiPoly Documented declaration: PhiPoly line 48 source
Source documentation
The "cleared denominator" polynomial: for n = P.natDegree, the rational function (x, y) ↦ (2xy−1)^(n+1) * Qfun P x y ((x+y)/(2xy−1)) is actually a polynomial function, and this is the corresponding polynomial, written as a polynomial in y whose coefficients are polynomials in x.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommRing, Polynomial
Proof uses: Finset.sum, Nat, Finset.range, HAdd.hAdd, Polynomial.natDegree, OfNat.ofNat, HMul.hMul, RingHom, Polynomial.C, Finsupp, Polynomial.coeff, HSub.hSub, HPow.hPow, Polynomial.X
36 instances, coercions and auxiliary constants left out.
```
noncomputable def PhiPoly {R : Type*} [CommRing R] (P : Polynomial R) :
    Polynomial (Polynomial R) :=
…
```
Mentions: P
lemma Usa2019P6.qfun_eq_sum Qfun expanded as a sum over the coefficients of P line 61 source
Source documentation
Qfun expanded as a sum over the coefficients of P.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Field, Polynomial, Eq, Usa2019P6.Qfun, Finset.sum, Nat, Finset.range, HAdd.hAdd, Polynomial.natDegree, OfNat.ofNat, HMul.hMul, Finsupp, Polynomial.coeff, HSub.hSub, HPow.hPow
Proof uses: Eq.mpr, Polynomial.eval, id, congrArg, Usa2019P6.Qfun.eq_1, congrFun', Eq.trans, congr, Polynomial.eval_eq_sum_range, Finset.mul_sum, Finset.sum_congr, rfl, Finset, Membership.mem, Mathlib.Tactic.Ring.of_eq, Nat.rawCast, Int.rawCast, inferInstance, Ring, Int.negOfNat and 39 more
84 instances, coercions and auxiliary constants left out.

Tactic trace: 5 steps
Open to load the checked before-and-after goal states.
```
lemma qfun_eq_sum {K : Type*} [Field K] (P : Polynomial K) (x y z : K) :
    Qfun P x y z = ∑ k ∈ Finset.range (P.natDegree + 1), P.coeff k *
      (x * x ^ k + y * y ^ k + z * z ^ k -
        x * y * z * ((x - y) ^ k + (y - z) ^ k + (z - x) ^ k)) := by
…
```
Mentions: P, Usa2019P6.Qfun
lemma Usa2019P6.phiPoly_eval Evaluating PhiPoly at a pair (x, y) with 2xy ≠ 1 gives the cleared value of Qfun at the surface point (x, y, (x+y)/(2xy−1)) line 74 source
Source documentation
Evaluating PhiPoly at a pair (x, y) with 2xy ≠ 1 gives the cleared value of Qfun at the surface point (x, y, (x+y)/(2xy−1)).

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Field, Polynomial, Ne, HMul.hMul, OfNat.ofNat, Nat, Eq, Polynomial.eval, Polynomial.map, Polynomial.evalRingHom, Usa2019P6.PhiPoly, HPow.hPow, HSub.hSub, HAdd.hAdd, Polynomial.natDegree, Usa2019P6.Qfun, HDiv.hDiv
Proof uses: Iff.mpr, sub_ne_zero, Eq.mpr, Polynomial.eval₂, id, congrArg, Polynomial.eval_map, Finset.sum, Finset.range, RingHom, Polynomial.C, Finsupp, Polynomial.coeff, Polynomial.X, Usa2019P6.PhiPoly.eq_1, Polynomial.eval₂_finsetSum, Usa2019P6.qfun_eq_sum, Finset.mul_sum, Finset.sum_congr, rfl and 107 more
144 instances, coercions and auxiliary constants left out.

Tactic trace: 22 steps
Open to load the checked before-and-after goal states.
```
lemma phiPoly_eval {K : Type*} [Field K] (P : Polynomial K) (x y : K)
    (hxy : 2 * x * y ≠ 1) :
    ((PhiPoly P).map (evalRingHom x)).eval y =
      (2 * x * y - 1) ^ (P.natDegree + 1) *
        Qfun P x y ((x + y) / (2 * x * y - 1)) := by
…
```
Mentions: P, Usa2019P6.PhiPoly, Usa2019P6.Qfun
lemma Usa2019P6.phiPoly_eq_zero If Qfun P vanishes at every nonzero real triple on the surface 2xyz = x+y+z, then PhiPoly P is the zero polynomial line 124 source
Source documentation
If Qfun P vanishes at every nonzero real triple on the surface 2xyz = x+y+z, then PhiPoly P is the zero polynomial. This is the "real dimension two" argument: a bivariate polynomial vanishing on a cofinite union of points is zero.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Polynomial, Real, Ne, OfNat.ofNat, Eq, HMul.hMul, Nat, HAdd.hAdd, Usa2019P6.Qfun, Usa2019P6.PhiPoly
Proof uses: Polynomial.map, Polynomial.evalRingHom, Polynomial.eq_zero_of_infinite_isRoot, LE.le, Set, Compl.compl, Finset, Insert.insert, Neg.neg, Singleton.singleton, Inv.inv, Set.ofPred, Polynomial.IsRoot, Membership.mem, of_eq_true, Eq.trans, Or, True, congr, congrArg and 129 more
165 instances, coercions and auxiliary constants left out.

Tactic trace: 46 steps
Open to load the checked before-and-after goal states.
```
lemma phiPoly_eq_zero (P : Polynomial ℝ)
    (hQ : ∀ x y z : ℝ, x ≠ 0 → y ≠ 0 → z ≠ 0 → 2 * x * y * z = x + y + z →
      Qfun P x y z = 0) :
    PhiPoly P = 0 := by
…
```
Mentions: P, Usa2019P6.Qfun, Usa2019P6.PhiPoly
lemma Usa2019P6.comp_neg_X_of_phiPoly_eq_zero P is even: from PhiPoly P = 0 we get Qfun P t 0 (-t) = 0 for all real t, hence t * (P(t) − P(−t)) = 0 for all t, so P.comp (-X) = P line 179 source
Source documentation
P is even: from PhiPoly P = 0 we get Qfun P t 0 (-t) = 0 for all real t, hence t * (P(t) − P(−t)) = 0 for all t, so P.comp (-X) = P.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Polynomial, Real, Eq, Usa2019P6.PhiPoly, OfNat.ofNat, Polynomial.comp, Neg.neg, Polynomial.X
Proof uses: Polynomial.eval, HMul.hMul, HSub.hSub, Polynomial.map, Polynomial.evalRingHom, Eq.mpr, id, congrArg, of_eq_true, Eq.trans, True, congrFun', Polynomial.map_zero, Polynomial.eval_zero, eq_self, Usa2019P6.Qfun, Nat, Mathlib.Tactic.Ring.of_eq, HAdd.hAdd, Int.rawCast and 93 more
118 instances, coercions and auxiliary constants left out.

Tactic trace: 24 steps
Open to load the checked before-and-after goal states.
```
lemma comp_neg_X_of_phiPoly_eq_zero (P : Polynomial ℝ) (hPhi : PhiPoly P = 0) :
    P.comp (-X) = P := by
…
```
Mentions: P, Usa2019P6.PhiPoly
lemma Usa2019P6.Fk_natDegree_le The second finite difference (X+h)^k + (X−h)^k − 2X^k has degree at most k−2 (in natural subtraction) line 203 source
Source documentation
The second finite difference (X+h)^k + (X−h)^k − 2X^k has degree at most k−2 (in natural subtraction). Expanded via the binomial theorem: the top two terms cancel.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Complex, Nat, LE.le, Polynomial.natDegree, HSub.hSub, Polynomial, HAdd.hAdd, HPow.hPow, Polynomial.X, RingHom, Polynomial.C, HMul.hMul, OfNat.ofNat
Proof uses: Nat.casesAuxOn, Eq, Mathlib.Tactic.Ring.of_eq, Mathlib.Tactic.Ring.Common.sub_congr, Nat.rawCast, Mathlib.Tactic.Ring.Common.add_congr, Mathlib.Tactic.Ring.Common.pow_congr, Mathlib.Tactic.Ring.Common.atom_pf, rfl, Eq.mpr, id, congrArg, Eq.symm, Eq.refl, Mathlib.Tactic.Ring.Common.add_pf_add_lt, Mathlib.Tactic.Ring.Common.add_pf_zero_add, Mathlib.Tactic.Ring.cast_zero, Mathlib.Meta.NormNum.isNat_ofNat, Mathlib.Tactic.Ring.Common.pow_zero, Int.rawCast and 77 more
133 instances, coercions and auxiliary constants left out.

Tactic trace: 33 steps
Open to load the checked before-and-after goal states.
```
lemma Fk_natDegree_le (h : ℂ) (k : ℕ) :
    ((X + C h) ^ k + (X - C h) ^ k - 2 * X ^ k : Polynomial ℂ).natDegree ≤ k - 2 := by
…
```
Mentions: h
lemma Usa2019P6.Fk_coeff The leading coefficient of the second finite difference (X+h)^k + (X−h)^k − 2X^k at degree k−2 is k * (k−1) * h² line 247 source
Source documentation
The leading coefficient of the second finite difference (X+h)^k + (X−h)^k − 2X^k at degree k−2 is k * (k−1) * h².

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Complex, Nat, Eq, Finsupp, Polynomial.coeff, HSub.hSub, Polynomial, HAdd.hAdd, HPow.hPow, Polynomial.X, RingHom, Polynomial.C, HMul.hMul, OfNat.ofNat
Proof uses: Nat.casesAuxOn, Mathlib.Tactic.Ring.of_eq, Mathlib.Tactic.Ring.Common.sub_congr, Nat.rawCast, Mathlib.Tactic.Ring.Common.add_congr, Mathlib.Tactic.Ring.Common.pow_congr, Mathlib.Tactic.Ring.Common.atom_pf, rfl, Eq.mpr, id, congrArg, Eq.symm, Eq.refl, Mathlib.Tactic.Ring.Common.add_pf_add_lt, Mathlib.Tactic.Ring.Common.add_pf_zero_add, Mathlib.Tactic.Ring.cast_zero, Mathlib.Meta.NormNum.isNat_ofNat, Mathlib.Tactic.Ring.Common.pow_zero, Int.rawCast, inferInstance and 129 more
185 instances, coercions and auxiliary constants left out.

Tactic trace: 50 steps
Open to load the checked before-and-after goal states.
```
lemma Fk_coeff (h : ℂ) (k : ℕ) :
    ((X + C h) ^ k + (X - C h) ^ k - 2 * X ^ k : Polynomial ℂ).coeff (k - 2) =
      (k : ℂ) * ((k : ℂ) - 1) * h ^ 2 := by
…
```
Mentions: h
lemma Usa2019P6.phiPoly_map_ofReal PhiPoly commutes with mapping the reals into the complexes line 308 source
Source documentation
PhiPoly commutes with mapping the reals into the complexes.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Polynomial, Real, Eq, Complex, Polynomial.map, Polynomial.mapRingHom, Complex.ofRealHom, Usa2019P6.PhiPoly
Proof uses: Function.Injective, Complex.ofReal_injective, Eq.mpr, Finset.sum, Nat, Finset.range, HAdd.hAdd, Polynomial.natDegree, OfNat.ofNat, HMul.hMul, RingHom, Polynomial.C, Finsupp, Polynomial.coeff, HSub.hSub, HPow.hPow, Polynomial.X, id, congrArg, Usa2019P6.PhiPoly.eq_1 and 25 more
59 instances, coercions and auxiliary constants left out.

Tactic trace: 5 steps
Open to load the checked before-and-after goal states.
```
lemma phiPoly_map_ofReal (P : Polynomial ℝ) :
    (PhiPoly P).map (mapRingHom Complex.ofRealHom) =
      PhiPoly (P.map Complex.ofRealHom) := by
…
```
Mentions: P, Usa2019P6.PhiPoly
lemma Usa2019P6.natDegree_le_two The degree bound: if PhiPoly P = 0 and P is even, then P.natDegree ≤ 2 line 323 source
Source documentation
The degree bound: if PhiPoly P = 0 and P is even, then P.natDegree ≤ 2. The proof passes to complex polynomials and evaluates Qfun at the surface point (x, h, -h) with h² = −1/2, obtaining that the second finite difference of P at h is constant; comparing leading coefficients then forces deg P ≤ 2.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Polynomial, Real, Ne, OfNat.ofNat, Eq, Usa2019P6.PhiPoly, Polynomial.comp, Neg.neg, Polynomial.X, LE.le, Nat, Polynomial.natDegree
Proof uses: Complex, HPow.hPow, HDiv.hDiv, Exists, Exists.intro, HMul.hMul, Complex.I, Real.sqrt, Eq.mpr, id, congrArg, mul_pow, Complex.I_sq, Eq.symm, Complex.ofReal_pow, div_pow, Real.sq_sqrt, zero_le_two, congrFun', Eq.trans and 217 more
211 instances, coercions and auxiliary constants left out.

Tactic trace: 140 steps
Open to load the checked before-and-after goal states.
```
lemma natDegree_le_two (P : Polynomial ℝ) (hP0 : P ≠ 0) (hPhi : PhiPoly P = 0)
    (hEven : P.comp (-X) = P) :
    P.natDegree ≤ 2 := by
…
```
Mentions: P, Usa2019P6.PhiPoly
problem Usa2019P6.usa2019_p6 line 472 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Polynomial, Real, Iff, Membership.mem, Set, Usa2019P6.SolutionSet, Ne, OfNat.ofNat, Eq, HMul.hMul, Nat, HAdd.hAdd, HDiv.hDiv, Polynomial.eval, HSub.hSub
Proof uses: Iff.intro, RingHom, Polynomial.C, HPow.hPow, Polynomial.X, Eq.mpr, id, congr, congrArg, congrFun', Eq.trans, Polynomial.eval_mul, Polynomial.eval_C, Polynomial.eval_add, Polynomial.eval_pow, Polynomial.eval_X, Polynomial.eval_ofNat, inferInstance, CommGroupWithZero, GroupWithZero and 223 more
194 instances, coercions and auxiliary constants left out.

Tactic trace: 56 steps
Open to load the checked before-and-after goal states.