The 14th Whitney Problems Workshop and Sobolev functions on subsets of β SOLUTIONS OF SEMIALGEBRAIC EQUATIONS Joint work with E. BIERSTONE and P.D. MILMAN Jean-Baptiste Campesato August 19, 2021 J.-B. Campesato (joint work with E. Bierstone and P.D. Milman) solutions of semialgebraic equations 1 / 20
59
Embed
The 14th Whitney Problems Workshop π and Sobolev functions ...
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
The 14th Whitney Problems Workshopππ and Sobolev functions on subsets of βπ
ππ SOLUTIONS OF SEMIALGEBRAIC EQUATIONS
Joint work with E. BIERSTONE and P.D. MILMAN
Jean-Baptiste Campesato
August 19, 2021
J.-B. Campesato (joint work with E. Bierstone and P.D. Milman) ππ solutions of semialgebraic equations 1 / 20
Semialgebraic geometry The problems The results The proof
Semialgebraic geometry β Definitions
Definition: semialgebraic setsSemialgebraic subsets of βπ are elements of the boolean algebra spanned by sets of the form
{π₯ β βπ βΆ π(π₯) β₯ 0}
where π β β[π₯1, β¦ , π₯π].
RemarkGiven π β β[π₯1, β¦ , π₯π], the following sets are semialgebraic
Definition: semialgebraic functionsLet π β βπ. A function π βΆ π β βπ is semialgebraic if its graph Ξπ β βπ+π is semialgebraic.
J.-B. Campesato (joint work with E. Bierstone and P.D. Milman) ππ solutions of semialgebraic equations 2 / 20
Semialgebraic geometry The problems The results The proof
Semialgebraic geometry β Definitions
Definition: semialgebraic setsSemialgebraic subsets of βπ are elements of the boolean algebra spanned by sets of the form
{π₯ β βπ βΆ π(π₯) β₯ 0}
where π β β[π₯1, β¦ , π₯π].
RemarkGiven π β β[π₯1, β¦ , π₯π], the following sets are semialgebraic
Definition: semialgebraic functionsLet π β βπ. A function π βΆ π β βπ is semialgebraic if its graph Ξπ β βπ+π is semialgebraic.
J.-B. Campesato (joint work with E. Bierstone and P.D. Milman) ππ solutions of semialgebraic equations 2 / 20
Semialgebraic geometry The problems The results The proof
Semialgebraic geometry β Definitions
Definition: semialgebraic setsSemialgebraic subsets of βπ are elements of the boolean algebra spanned by sets of the form
{π₯ β βπ βΆ π(π₯) β₯ 0}
where π β β[π₯1, β¦ , π₯π].
RemarkGiven π β β[π₯1, β¦ , π₯π], the following sets are semialgebraic
Definition: semialgebraic functionsLet π β βπ. A function π βΆ π β βπ is semialgebraic if its graph Ξπ β βπ+π is semialgebraic.
J.-B. Campesato (joint work with E. Bierstone and P.D. Milman) ππ solutions of semialgebraic equations 2 / 20
Semialgebraic geometry The problems The results The proof
Semialgebraic geometry β Definitions
Definition: semialgebraic setsSemialgebraic subsets of βπ are elements of the boolean algebra spanned by sets of the form
{π₯ β βπ βΆ π(π₯) β₯ 0}
where π β β[π₯1, β¦ , π₯π].
RemarkGiven π β β[π₯1, β¦ , π₯π], the following sets are semialgebraic
Definition: semialgebraic functionsLet π β βπ. A function π βΆ π β βπ is semialgebraic if its graph Ξπ β βπ+π is semialgebraic.
J.-B. Campesato (joint work with E. Bierstone and P.D. Milman) ππ solutions of semialgebraic equations 2 / 20
Semialgebraic geometry The problems The results The proof
Semialgebraic geometry β TarskiβSeidenberg theoremTheorem (TarskiβSeidenberg): semialgebraic sets are closed under projections
If π β βπ+1 is semialgebraic then so is π(π), where π βΆ βπ+1 β βπ, π(π₯1, β¦ , π₯π+1) = (π₯1, β¦ , π₯π).
RemarkIf π βΆ π β βπ is semialgebraic then so is π.
Corollary: elimination of quantifiers
Let π β βπ+1 be semialgebraic, then the following sets are too
J.-B. Campesato (joint work with E. Bierstone and P.D. Milman) ππ solutions of semialgebraic equations 3 / 20
Semialgebraic geometry The problems The results The proof
Semialgebraic geometry β TarskiβSeidenberg theoremTheorem (TarskiβSeidenberg): semialgebraic sets are closed under projections
If π β βπ+1 is semialgebraic then so is π(π), where π βΆ βπ+1 β βπ, π(π₯1, β¦ , π₯π+1) = (π₯1, β¦ , π₯π).
RemarkIf π βΆ π β βπ is semialgebraic then so is π.
Corollary: elimination of quantifiers
Let π β βπ+1 be semialgebraic, then the following sets are too
J.-B. Campesato (joint work with E. Bierstone and P.D. Milman) ππ solutions of semialgebraic equations 3 / 20
Semialgebraic geometry The problems The results The proof
Semialgebraic geometry β TarskiβSeidenberg theoremTheorem (TarskiβSeidenberg): semialgebraic sets are closed under projections
If π β βπ+1 is semialgebraic then so is π(π), where π βΆ βπ+1 β βπ, π(π₯1, β¦ , π₯π+1) = (π₯1, β¦ , π₯π).
RemarkIf π βΆ π β βπ is semialgebraic then so is π.
Corollary: elimination of quantifiers
Let π β βπ+1 be semialgebraic, then the following sets are too
J.-B. Campesato (joint work with E. Bierstone and P.D. Milman) ππ solutions of semialgebraic equations 3 / 20
Semialgebraic geometry The problems The results The proof
Semialgebraic geometry β TarskiβSeidenberg theoremTheorem (TarskiβSeidenberg): semialgebraic sets are closed under projections
If π β βπ+1 is semialgebraic then so is π(π), where π βΆ βπ+1 β βπ, π(π₯1, β¦ , π₯π+1) = (π₯1, β¦ , π₯π).
RemarkIf π βΆ π β βπ is semialgebraic then so is π.
Corollary: elimination of quantifiers
Let π β βπ+1 be semialgebraic, then the following sets are too
there exists a ππ semialgebraic function πΉ βΆ βπ β β such that π·πΌ πΉ|π = ππΌ and πΉ is Nash on βπ ⧡ π.
Nash β semialgebraic and analytic.
J.-B. Campesato (joint work with E. Bierstone and P.D. Milman) ππ solutions of semialgebraic equations 4 / 20
Semialgebraic geometry The problems The results The proof
Are there solutions preserving semialgebraicity?
For Whitneyβs Extension Problem
Let π βΆ π β β be a semialgebraic function where π β βπ is closed.If π admits a ππ extension πΉ βΆ βπ β β, does it admit a semialgebraic ππ extension πΉ βΆ βπ β β?
For the BrennerβFeffermanβHochsterβKollΓ‘r Problem
Let π1, β¦ , ππ, π βΆ βπ β β be semialgebraic functions.If the equation π = β ππππ admit a ππ solution (ππ)π, does it admit a semialgebraic ππ solution?
β’ AschenbrennerβThamrongthanyalak (2019): βπ, for π = 1 and π = 0, respectively.β’ FeffermanβLuli (2021): βπ, for π = 2.
β’ BierstoneβC.βMilman (2021): βπ, βπ, with a loss of differentiability.
J.-B. Campesato (joint work with E. Bierstone and P.D. Milman) ππ solutions of semialgebraic equations 5 / 20
Semialgebraic geometry The problems The results The proof
Are there solutions preserving semialgebraicity?
For Whitneyβs Extension ProblemLet π βΆ π β β be a semialgebraic function where π β βπ is closed.If π admits a ππ extension πΉ βΆ βπ β β, does it admit a semialgebraic ππ extension πΉ βΆ βπ β β?
For the BrennerβFeffermanβHochsterβKollΓ‘r Problem
Let π1, β¦ , ππ, π βΆ βπ β β be semialgebraic functions.If the equation π = β ππππ admit a ππ solution (ππ)π, does it admit a semialgebraic ππ solution?
β’ AschenbrennerβThamrongthanyalak (2019): βπ, for π = 1 and π = 0, respectively.β’ FeffermanβLuli (2021): βπ, for π = 2.β’ BierstoneβC.βMilman (2021): βπ, βπ, with a loss of differentiability.
J.-B. Campesato (joint work with E. Bierstone and P.D. Milman) ππ solutions of semialgebraic equations 5 / 20
Semialgebraic geometry The problems The results The proof
Are there solutions preserving semialgebraicity?
For Whitneyβs Extension ProblemLet π βΆ π β β be a semialgebraic function where π β βπ is closed.If π admits a ππ extension πΉ βΆ βπ β β, does it admit a semialgebraic ππ extension πΉ βΆ βπ β β?
For the BrennerβFeffermanβHochsterβKollΓ‘r ProblemLet π1, β¦ , ππ, π βΆ βπ β β be semialgebraic functions.If the equation π = β ππππ admit a ππ solution (ππ)π, does it admit a semialgebraic ππ solution?
β’ AschenbrennerβThamrongthanyalak (2019): βπ, for π = 1 and π = 0, respectively.β’ FeffermanβLuli (2021): βπ, for π = 2.β’ BierstoneβC.βMilman (2021): βπ, βπ, with a loss of differentiability.
J.-B. Campesato (joint work with E. Bierstone and P.D. Milman) ππ solutions of semialgebraic equations 5 / 20
Semialgebraic geometry The problems The results The proof
Are there solutions preserving semialgebraicity?
For Whitneyβs Extension ProblemLet π βΆ π β β be a semialgebraic function where π β βπ is closed.If π admits a ππ extension πΉ βΆ βπ β β, does it admit a semialgebraic ππ extension πΉ βΆ βπ β β?
For the BrennerβFeffermanβHochsterβKollΓ‘r ProblemLet π1, β¦ , ππ, π βΆ βπ β β be semialgebraic functions.If the equation π = β ππππ admit a ππ solution (ππ)π, does it admit a semialgebraic ππ solution?
β’ AschenbrennerβThamrongthanyalak (2019): βπ, for π = 1 and π = 0, respectively.β’ FeffermanβLuli (2021): βπ, for π = 2.β’ BierstoneβC.βMilman (2021): βπ, βπ, with a loss of differentiability.
J.-B. Campesato (joint work with E. Bierstone and P.D. Milman) ππ solutions of semialgebraic equations 5 / 20
Semialgebraic geometry The problems The results The proof
Strategy for the semialgebraic π1 extension problem(AschenbrennerβThamrongthanyalak, 2019)
Let π βΆ π β β be a semialgebraic function admitting a π1 extension πΉ βΆ βπ β β.
β’ Then π is semialgebraic, and, βπ₯ β π, (π₯, πΉ (π₯), βπΉ (π₯)) β π.β’ Semialgebraic Michaelβs Selection Lemma:
there exists π βΆ π β π semialgebraic and continuous such that ππ₯ β π = id where ππ₯(π₯, π¦, π£) = π₯.β’ Set πΊ β ππ£ β π βΆ βπ β βπ where ππ£(π₯, π¦, π£) = π£, then πΊ is semialgebraic, continuous and satisfies
β’ Then π is semialgebraic, and, βπ₯ β π, (π₯, πΉ (π₯), βπΉ (π₯)) β π.β’ Semialgebraic Michaelβs Selection Lemma:
there exists π βΆ π β π semialgebraic and continuous such that ππ₯ β π = id where ππ₯(π₯, π¦, π£) = π₯.β’ Set πΊ β ππ£ β π βΆ βπ β βπ where ππ£(π₯, π¦, π£) = π£, then πΊ is semialgebraic, continuous and satisfies
there exists π βΆ π β π semialgebraic and continuous such that ππ₯ β π = id where ππ₯(π₯, π¦, π£) = π₯.β’ Set πΊ β ππ£ β π βΆ βπ β βπ where ππ£(π₯, π¦, π£) = π£, then πΊ is semialgebraic, continuous and satisfies
β’ Then π is semialgebraic, and, βπ₯ β π, (π₯, πΉ (π₯), βπΉ (π₯)) β π.β’ Semialgebraic Michaelβs Selection Lemma:
there exists π βΆ π β π semialgebraic and continuous such that ππ₯ β π = id where ππ₯(π₯, π¦, π£) = π₯.β’ Set πΊ β ππ£ β π βΆ βπ β βπ where ππ£(π₯, π¦, π£) = π£, then πΊ is semialgebraic, continuous and satisfies
β’ Then π is semialgebraic, and, βπ₯ β π, (π₯, πΉ (π₯), βπΉ (π₯)) β π.β’ Semialgebraic Michaelβs Selection Lemma:
there exists π βΆ π β π semialgebraic and continuous such that ππ₯ β π = id where ππ₯(π₯, π¦, π£) = π₯.β’ Set πΊ β ππ£ β π βΆ βπ β βπ where ππ£(π₯, π¦, π£) = π£, then πΊ is semialgebraic, continuous and satisfies
2 According to the definable choice: there exist π Β±π semialgebraic satisfying (β).
3 Then πΉ (π₯, π¦) = πβ(π₯, π¦)(
π
βπ=0
π βπ (π₯)π! π¦π
)+ π+(π₯, π¦)
(
π
βπ=0
π +π (π₯)π! (π¦ β π(π₯))π
)is a semialgebraic ππ
extension of π in a neighborhood of the origin such that π½(0,0) πΉ = 0.
J.-B. Campesato (joint work with E. Bierstone and P.D. Milman) ππ solutions of semialgebraic equations 7 / 20
Semialgebraic geometry The problems The results The proof
The main results: statements
Theorem β BierstoneβC.βMilman, 2021Given π β βπ closed and semialgebraic, there exists π βΆ β β β satisfying the following property:if π βΆ π β β semialgebraic admits a ππ(π) extension, then it admits a semialgebraic ππ extension.
Theorem β BierstoneβC.βMilman, 2021Given π΄ βΆ βπ β β³π,π(β) semialgebraic, there exists π βΆ β β β such that:if πΉ βΆ βπ β βπ semialgebraic may be written πΉ (π₯) = π΄(π₯)πΊ(π₯) where πΊ is ππ(π),then πΉ (π₯) = π΄(π₯)οΏ½οΏ½(π₯) where οΏ½οΏ½(π₯) is semialgebraic and ππ.
J.-B. Campesato (joint work with E. Bierstone and P.D. Milman) ππ solutions of semialgebraic equations 8 / 20
Semialgebraic geometry The problems The results The proof
The main results: statements
Theorem β BierstoneβC.βMilman, 2021Given π β βπ closed and semialgebraic, there exists π βΆ β β β satisfying the following property:if π βΆ π β β semialgebraic admits a ππ(π) extension, then it admits a semialgebraic ππ extension.
Theorem β BierstoneβC.βMilman, 2021Given π΄ βΆ βπ β β³π,π(β) semialgebraic, there exists π βΆ β β β such that:if πΉ βΆ βπ β βπ semialgebraic may be written πΉ (π₯) = π΄(π₯)πΊ(π₯) where πΊ is ππ(π),then πΉ (π₯) = π΄(π₯)οΏ½οΏ½(π₯) where οΏ½οΏ½(π₯) is semialgebraic and ππ.
J.-B. Campesato (joint work with E. Bierstone and P.D. Milman) ππ solutions of semialgebraic equations 8 / 20
Semialgebraic geometry The problems The results The proof
Main results: towards a common generalization
The extension problemLet π β βπ be semialgebraic and closed.
By resolution of singularities, there exists π βΆ π β βπ
Nash and proper defined on a Nash manifold suchthat π = π(π).
Given π βΆ βπ β β and π βΆ π β β, we haveπ|π = π if and only if
where π΄ β π΄ β π is now Nash and πΉ β πΉ β π.
J.-B. Campesato (joint work with E. Bierstone and P.D. Milman) ππ solutions of semialgebraic equations 9 / 20
Semialgebraic geometry The problems The results The proof
The main result
Theorem β BierstoneβC.βMilman, 2021Let π΄ βΆ βπ β β³π,π(β) be Nash and let π βΆ π β βπ be Nash and proper defined on π β βπ aNash submanifold.Then there exists π βΆ β β β satisfying the following property.If π βΆ π β βπ semialgebraic may be written
π(π₯) = π΄(π₯)π(π(π₯))
for a ππ(π) function π βΆ βπ β βπ then
π(π₯) = π΄(π₯) π(π(π₯))
for a semialgebraic ππ function π.
J.-B. Campesato (joint work with E. Bierstone and P.D. Milman) ππ solutions of semialgebraic equations 10 / 20
Semialgebraic geometry The problems The results The proof
Heart of the proof: induction on dimension
Proposition: the induction stepLet π΅ β π(π) be semialgebraic and closed.There exist π΅β² β π΅ semialgebraic satisfying dim π΅β² < dim π΅ and π‘ βΆ β β β such that if
1 π βΆ π β βπ is ππ‘(π), semialgebraic and π‘(π)-flat on πβ1(π΅β²), and2 π = π΄ β (π β π) admits a ππ‘(π) solution π,
then there exists a semialgebraic ππ function π βΆ βπ β βπ s.t. π β π΄ β ( π β π) is π-flat on πβ1(π΅).
Then, up to subtracting by π΄ β ( π β π) on both side, we get an equation
π = π΄ β (π β π)
where π is now semialgebraic and π-flat on πβ1(π΅).
J.-B. Campesato (joint work with E. Bierstone and P.D. Milman) ππ solutions of semialgebraic equations 11 / 20
Semialgebraic geometry The problems The results The proof
Heart of the proof: induction on dimension
Proposition: the induction stepLet π΅ β π(π) be semialgebraic and closed.There exist π΅β² β π΅ semialgebraic satisfying dim π΅β² < dim π΅ and π‘ βΆ β β β such that if
1 π βΆ π β βπ is ππ‘(π), semialgebraic and π‘(π)-flat on πβ1(π΅β²), and2 π = π΄ β (π β π) admits a ππ‘(π) solution π,
then there exists a semialgebraic ππ function π βΆ βπ β βπ s.t. π β π΄ β ( π β π) is π-flat on πβ1(π΅).
Then, up to subtracting by π΄ β ( π β π) on both side, we get an equation
π = π΄ β (π β π)
where π is now semialgebraic and π-flat on πβ1(π΅).
J.-B. Campesato (joint work with E. Bierstone and P.D. Milman) ππ solutions of semialgebraic equations 11 / 20
Semialgebraic geometry The problems The results The proof
Heart of the proof: induction on dimension
Strategy: construction of a semialgebraic Whitney field
ππ΄(x) πΊ(π, π πππ(x)) mod (x)π+1βJxKπ
A - Whitney regularityGiven π΅, there exists π β β such that if πΊ is a Whitney field of order π β₯ ππ on π΅ ⧡ π΅β² then it is aWhitney field of order π on π΅.
J.-B. Campesato (joint work with E. Bierstone and P.D. Milman) ππ solutions of semialgebraic equations 12 / 20
Semialgebraic geometry The problems The results The proof
Heart of the proof: induction on dimension
Strategy: construction of a semialgebraic Whitney field
ππ΄(x) πΊ(π, π πππ(x)) mod (x)π+1βJxKπ
A - Whitney regularityGiven π΅, there exists π β β such that if πΊ is a Whitney field of order π β₯ ππ on π΅ ⧡ π΅β² then it is aWhitney field of order π on π΅.
J.-B. Campesato (joint work with E. Bierstone and P.D. Milman) ππ solutions of semialgebraic equations 12 / 20
Semialgebraic geometry The problems The results The proof
The module of relations at π β π(π)We consider the equation at the level of Taylor polynomials:
π ππ π(x) β‘ π π
π π΄(x) πΊ(π, π ππ π(x)) mod (x)π+1βJxKπ (1)
B - Chevalleyβs functionGiven π β β, there exists π β₯ π such that the derivatives of π of order β€ π can be expressed interms of the derivatives of π of order β€ π.
Formally, we stratify π΅ = β¨πmaxπ=1 Ξπ such that for each π, there exists π β₯ π satisfying
whereβ’ βπ(π) is the module of relations at π β π(π) formed by the πΊ β βJyKπ satisfying the
homogeneous equation associated to (1) for all π β πβ1(π), and,β’ ππ is the truncation up to degree π.
J.-B. Campesato (joint work with E. Bierstone and P.D. Milman) ππ solutions of semialgebraic equations 13 / 20
Semialgebraic geometry The problems The results The proof
The module of relations at π β π(π)We consider the equation at the level of Taylor polynomials:
π ππ π(x) β‘ π π
π π΄(x) πΊ(π, π ππ π(x)) mod (x)π+1βJxKπ (1)
B - Chevalleyβs functionGiven π β β, there exists π β₯ π such that the derivatives of π of order β€ π can be expressed interms of the derivatives of π of order β€ π.
Formally, we stratify π΅ = β¨πmaxπ=1 Ξπ such that for each π, there exists π β₯ π satisfying
whereβ’ βπ(π) is the module of relations at π β π(π) formed by the πΊ β βJyKπ satisfying the
homogeneous equation associated to (1) for all π β πβ1(π), and,β’ ππ is the truncation up to degree π.
J.-B. Campesato (joint work with E. Bierstone and P.D. Milman) ππ solutions of semialgebraic equations 13 / 20
Semialgebraic geometry The problems The results The proof
The module of relations at π β π(π)We consider the equation at the level of Taylor polynomials:
π ππ π(x) β‘ π π
π π΄(x) πΊ(π, π ππ π(x)) mod (x)π+1βJxKπ (1)
B - Chevalleyβs functionGiven π β β, there exists π β₯ π such that the derivatives of π of order β€ π can be expressed interms of the derivatives of π of order β€ π.
Formally, we stratify π΅ = β¨πmaxπ=1 Ξπ such that for each π, there exists π β₯ π satisfying
J.-B. Campesato (joint work with E. Bierstone and P.D. Milman) ππ solutions of semialgebraic equations 15 / 20
Semialgebraic geometry The problems The results The proof
Diagram of initial exponents and module of relations
Lemma β Chevalleyβs functionLet π β β.There exists (Ξπ)π a stratification of π΅ such that given a stratum Ξπ , there exists π β₯ π satisfying
J.-B. Campesato (joint work with E. Bierstone and P.D. Milman) ππ solutions of semialgebraic equations 16 / 20
Semialgebraic geometry The problems The results The proof
Diagram of initial exponents and module of relations
Lemma β Chevalleyβs functionLet π β β.There exists (Ξπ)π a stratification of π΅ such that given a stratum Ξπ , there exists π β₯ π satisfying
J.-B. Campesato (joint work with E. Bierstone and P.D. Milman) ππ solutions of semialgebraic equations 16 / 20
Semialgebraic geometry The problems The results The proof
Construction of πΊ on Ξπ
For π β π΅ and π‘ β₯ π, by assumption there exists ππ β β[y]π such that
π π‘ππ(x) β‘ π π‘
ππ΄(x) ππ(π π‘ππ(x)) mod (x)π‘+1βJxKπ, βπ β πβ1(π).
Letβs fix a stratum Ξπ and π β Ξπ .By formal division, we may write ππ(y) = β ππ(y)Ξ¦π(y) + ππ(π, y) where the Ξ¦π are as above for βπ(π).Note that the remainder ππ(π, y) β β[y]π satisfies
Semialgebraic geometry The problems The results The proof
Construction of πΊ on Ξπ
For π β π΅ and π‘ β₯ π, by assumption there exists ππ β β[y]π such that
π π‘ππ(x) β‘ π π‘
ππ΄(x) ππ(π π‘ππ(x)) mod (x)π‘+1βJxKπ, βπ β πβ1(π).
Letβs fix a stratum Ξπ and π β Ξπ .By formal division, we may write ππ(y) = β ππ(y)Ξ¦π(y) + ππ(π, y) where the Ξ¦π are as above for βπ(π).Note that the remainder ππ(π, y) β β[y]π satisfies
Semialgebraic geometry The problems The results The proof
Construction of πΊ on Ξπ
For π β π΅ and π‘ β₯ π, by assumption there exists ππ β β[y]π such that
π π‘ππ(x) β‘ π π‘
ππ΄(x) ππ(π π‘ππ(x)) mod (x)π‘+1βJxKπ, βπ β πβ1(π).
Letβs fix a stratum Ξπ and π β Ξπ .By formal division, we may write ππ(y) = β ππ(y)Ξ¦π(y) + ππ(π, y) where the Ξ¦π are as above for βπ(π).Note that the remainder ππ(π, y) β β[y]π satisfies
Semialgebraic geometry The problems The results The proof
πΊ is a Whitney field of order π on ΞπTo simplify the situation, we omit π.Thanks to Borelβs lemma with parameter, it is enough to check that π·π,π£πΊπβ1
J.-B. Campesato (joint work with E. Bierstone and P.D. Milman) ππ solutions of semialgebraic equations 18 / 20
Semialgebraic geometry The problems The results The proof
πΊ is a Whitney field of order π on ΞπTo simplify the situation, we omit π.Thanks to Borelβs lemma with parameter, it is enough to check that π·π,π£πΊπβ1
J.-B. Campesato (joint work with E. Bierstone and P.D. Milman) ππ solutions of semialgebraic equations 18 / 20
Semialgebraic geometry The problems The results The proof
πΊ is a Whitney field of order π on ΞπTo simplify the situation, we omit π.Thanks to Borelβs lemma with parameter, it is enough to check that π·π,π£πΊπβ1
J.-B. Campesato (joint work with E. Bierstone and P.D. Milman) ππ solutions of semialgebraic equations 18 / 20
Semialgebraic geometry The problems The results The proof
πΊ is a Whitney field of order π on ΞπTo simplify the situation, we omit π.Thanks to Borelβs lemma with parameter, it is enough to check that π·π,π£πΊπβ1
J.-B. Campesato (joint work with E. Bierstone and P.D. Milman) ππ solutions of semialgebraic equations 18 / 20
Semialgebraic geometry The problems The results The proof
πΊ is a Whitney field of order π on ΞπTo simplify the situation, we omit π.Thanks to Borelβs lemma with parameter, it is enough to check that π·π,π£πΊπβ1
π π΄(x) πΊ(π, π ππ π(x)) mod (x)π+1βJxKπ
Hence we obtain π βΆ βπ β βπ semialgebraic and ππ such that π β π΄ β (π β π) is π-flat on πβ1(π΅).
Loss of differentiabilityFor π β β, we set π β₯ ππ, then π β₯ π(π) and finally π‘(π) β π‘ β₯ π + π whereA. π is an upper bound of Whitneyβs loss of differentiability (induction step).B. π βΆ β β β is an upper bound of the Chevalley functions on the various strata.C. π is an upper bound of Εojasiewiczβs loss of differentiability on each stratum.
J.-B. Campesato (joint work with E. Bierstone and P.D. Milman) ππ solutions of semialgebraic equations 20 / 20
Semialgebraic geometry The problems The results The proof
Summary
We constructed πΊ(π, y) = β|πΌ|β€π
ππΌ (π)πΌ! yπΌ a semialgebraic Whitney field of order π on π΅ such that
π π΄(x) πΊ(π, π ππ π(x)) mod (x)π+1βJxKπ
Hence we obtain π βΆ βπ β βπ semialgebraic and ππ such that π β π΄ β (π β π) is π-flat on πβ1(π΅).
Loss of differentiabilityFor π β β, we set π β₯ ππ, then π β₯ π(π) and finally π‘(π) β π‘ β₯ π + π whereA. π is an upper bound of Whitneyβs loss of differentiability (induction step).B. π βΆ β β β is an upper bound of the Chevalley functions on the various strata.C. π is an upper bound of Εojasiewiczβs loss of differentiability on each stratum.
J.-B. Campesato (joint work with E. Bierstone and P.D. Milman) ππ solutions of semialgebraic equations 20 / 20