Multivariable Puiseux Theorem for Convergent Generalised Power - - PowerPoint PPT Presentation

Multivariable Puiseux Theorem for Convergent Generalised Power Series

Tamara Servi (CMAF Lisboa)

Multivariable Puiseux Theorem for Convergent Generalised Power Series

Tamara Servi (CMAF Lisboa)

This talk is about solving equations in a class which extends that of real analytic functions.

Multivariable Puiseux Theorem for Convergent Generalised Power Series

Tamara Servi (CMAF Lisboa)

This talk is about solving equations in a class which extends that of real analytic functions. On = real analytic germs at 0 ∈ Rn

Multivariable Puiseux Theorem for Convergent Generalised Power Series

Tamara Servi (CMAF Lisboa)

This talk is about solving equations in a class which extends that of real analytic functions. On = real analytic germs at 0 ∈ Rn

• Puiseux. f (x, y) ∈ O2, f (0, 0) = 0 ⇒ the solutions y = ϕ (x) of f = 0 in a

nbd of 0 ∈ R2 are convergent Puiseux series y =

i∈N aixi/d (for some d ∈ N).

Multivariable Puiseux Theorem for Convergent Generalised Power Series

Tamara Servi (CMAF Lisboa)

This talk is about solving equations in a class which extends that of real analytic functions. On = real analytic germs at 0 ∈ Rn

• Puiseux. f (x, y) ∈ O2, f (0, 0) = 0 ⇒ the solutions y = ϕ (x) of f = 0 in a

nbd of 0 ∈ R2 are convergent Puiseux series y =

i∈N aixi/d (for some d ∈ N).

Multivariable version (vdDries-Marker-Macintyre, Lion-Rolin, Parusinski).

Multivariable Puiseux Theorem for Convergent Generalised Power Series

Tamara Servi (CMAF Lisboa)

This talk is about solving equations in a class which extends that of real analytic functions. On = real analytic germs at 0 ∈ Rn

• Puiseux. f (x, y) ∈ O2, f (0, 0) = 0 ⇒ the solutions y = ϕ (x) of f = 0 in a

nbd of 0 ∈ R2 are convergent Puiseux series y =

i∈N aixi/d (for some d ∈ N).

Multivariable version (vdDries-Marker-Macintyre, Lion-Rolin, Parusinski). x = (x1, . . . , xm) , f ∈ Om+1, f (0, 0) = 0 ⇒ the solutions y = ϕ (x) of f = 0 in a nbd of 0 ∈ Rm+1 are terms of the language underlying the following collection of germs:

Multivariable Puiseux Theorem for Convergent Generalised Power Series

Tamara Servi (CMAF Lisboa)

This talk is about solving equations in a class which extends that of real analytic functions. On = real analytic germs at 0 ∈ Rn

• Puiseux. f (x, y) ∈ O2, f (0, 0) = 0 ⇒ the solutions y = ϕ (x) of f = 0 in a

nbd of 0 ∈ R2 are convergent Puiseux series y =

i∈N aixi/d (for some d ∈ N).

Multivariable version (vdDries-Marker-Macintyre, Lion-Rolin, Parusinski). x = (x1, . . . , xm) , f ∈ Om+1, f (0, 0) = 0 ⇒ the solutions y = ϕ (x) of f = 0 in a nbd of 0 ∈ Rm+1 are terms of the language underlying the following collection of germs: A =

n∈N On ∪

• x →

d

√x : d ∈ N

• ∪ {x → 1/x}.

Multivariable Puiseux Theorem for Convergent Generalised Power Series

Tamara Servi (CMAF Lisboa)

This talk is about solving equations in a class which extends that of real analytic functions. On = real analytic germs at 0 ∈ Rn

• Puiseux. f (x, y) ∈ O2, f (0, 0) = 0 ⇒ the solutions y = ϕ (x) of f = 0 in a

nbd of 0 ∈ R2 are convergent Puiseux series y =

i∈N aixi/d (for some d ∈ N).

Multivariable version (vdDries-Marker-Macintyre, Lion-Rolin, Parusinski). x = (x1, . . . , xm) , f ∈ Om+1, f (0, 0) = 0 ⇒ the solutions y = ϕ (x) of f = 0 in a nbd of 0 ∈ Rm+1 are terms of the language underlying the following collection of germs: A =

n∈N On ∪

• x →

d

√x : d ∈ N

• ∪ {x → 1/x}.

Our goal: extend this result to a class of functions which generate an

• -minimal expansion of Ran:

Multivariable Puiseux Theorem for Convergent Generalised Power Series

Tamara Servi (CMAF Lisboa)

This talk is about solving equations in a class which extends that of real analytic functions. On = real analytic germs at 0 ∈ Rn

• Puiseux. f (x, y) ∈ O2, f (0, 0) = 0 ⇒ the solutions y = ϕ (x) of f = 0 in a

nbd of 0 ∈ R2 are convergent Puiseux series y =

i∈N aixi/d (for some d ∈ N).

Multivariable version (vdDries-Marker-Macintyre, Lion-Rolin, Parusinski). x = (x1, . . . , xm) , f ∈ Om+1, f (0, 0) = 0 ⇒ the solutions y = ϕ (x) of f = 0 in a nbd of 0 ∈ Rm+1 are terms of the language underlying the following collection of germs: A =

n∈N On ∪

• x →

d

√x : d ∈ N

• ∪ {x → 1/x}.

Our goal: extend this result to a class of functions which generate an

• -minimal expansion of Ran: Convergent Generalised Power Series.

The ring of formal generalised power series R x∗. Let x = (x1, . . . , xm).

The ring of formal generalised power series R x∗. Let x = (x1, . . . , xm). F (x) =

α cαxα such that α ∈ [0, ∞)m, cα ∈ R and

The ring of formal generalised power series R x∗. Let x = (x1, . . . , xm). F (x) =

α cαxα such that α ∈ [0, ∞)m, cα ∈ R and

Supp (F) := {α : cα = 0}

The ring of formal generalised power series R x∗. Let x = (x1, . . . , xm). F (x) =

α cαxα such that α ∈ [0, ∞)m, cα ∈ R and

Supp (F) := {α : cα = 0}⊆ S1 × . . . × Sm, where Si ⊆ [0, ∞) well ordered.

The ring of formal generalised power series R x∗. Let x = (x1, . . . , xm). F (x) =

α cαxα such that α ∈ [0, ∞)m, cα ∈ R and

Supp (F) := {α : cα = 0}⊆ S1 × . . . × Sm, where Si ⊆ [0, ∞) well ordered. (F has finitely many minimal monomials)

The ring of formal generalised power series R x∗. Let x = (x1, . . . , xm). F (x) =

α cαxα such that α ∈ [0, ∞)m, cα ∈ R and

Supp (F) := {α : cα = 0}⊆ S1 × . . . × Sm, where Si ⊆ [0, ∞) well ordered. (F has finitely many minimal monomials) F convergent if ∃r ∈ (0, ∞)m

α |cα|r |α| < ∞

The ring of formal generalised power series R x∗. Let x = (x1, . . . , xm). F (x) =

α cαxα such that α ∈ [0, ∞)m, cα ∈ R and

Supp (F) := {α : cα = 0}⊆ S1 × . . . × Sm, where Si ⊆ [0, ∞) well ordered. (F has finitely many minimal monomials) F convergent if ∃r ∈ (0, ∞)m

α |cα|r |α| < ∞ (sup of all finite subsums).

The ring of formal generalised power series R x∗. Let x = (x1, . . . , xm). F (x) =

α cαxα such that α ∈ [0, ∞)m, cα ∈ R and

Supp (F) := {α : cα = 0}⊆ S1 × . . . × Sm, where Si ⊆ [0, ∞) well ordered. (F has finitely many minimal monomials) F convergent if ∃r ∈ (0, ∞)m

α |cα|r |α| < ∞ (sup of all finite subsums).

F induces a C0 function (the sum of the series) on [0, r)m (analytic on (0, r)m).

The ring of formal generalised power series R x∗. Let x = (x1, . . . , xm). F (x) =

α cαxα such that α ∈ [0, ∞)m, cα ∈ R and

Supp (F) := {α : cα = 0}⊆ S1 × . . . × Sm, where Si ⊆ [0, ∞) well ordered. (F has finitely many minimal monomials) F convergent if ∃r ∈ (0, ∞)m

α |cα|r |α| < ∞ (sup of all finite subsums).

F induces a C0 function (the sum of the series) on [0, r)m (analytic on (0, r)m). R {x∗} =

r R {x∗}r is the ring of convergent generalised power series.

The ring of formal generalised power series R x∗. Let x = (x1, . . . , xm). F (x) =

α cαxα such that α ∈ [0, ∞)m, cα ∈ R and

Supp (F) := {α : cα = 0}⊆ S1 × . . . × Sm, where Si ⊆ [0, ∞) well ordered. (F has finitely many minimal monomials) F convergent if ∃r ∈ (0, ∞)m

α |cα|r |α| < ∞ (sup of all finite subsums).

F induces a C0 function (the sum of the series) on [0, r)m (analytic on (0, r)m). R {x∗} =

r R {x∗}r is the ring of convergent generalised power series.

Examples.

The ring of formal generalised power series R x∗. Let x = (x1, . . . , xm). F (x) =

α cαxα such that α ∈ [0, ∞)m, cα ∈ R and

Supp (F) := {α : cα = 0}⊆ S1 × . . . × Sm, where Si ⊆ [0, ∞) well ordered. (F has finitely many minimal monomials) F convergent if ∃r ∈ (0, ∞)m

α |cα|r |α| < ∞ (sup of all finite subsums).

F induces a C0 function (the sum of the series) on [0, r)m (analytic on (0, r)m). R {x∗} =

r R {x∗}r is the ring of convergent generalised power series.

Examples.

• f ∈ Om, αi ∈ [0, ∞);

The ring of formal generalised power series R x∗. Let x = (x1, . . . , xm). F (x) =

α cαxα such that α ∈ [0, ∞)m, cα ∈ R and

Supp (F) := {α : cα = 0}⊆ S1 × . . . × Sm, where Si ⊆ [0, ∞) well ordered. (F has finitely many minimal monomials) F convergent if ∃r ∈ (0, ∞)m

α |cα|r |α| < ∞ (sup of all finite subsums).

F induces a C0 function (the sum of the series) on [0, r)m (analytic on (0, r)m). R {x∗} =

r R {x∗}r is the ring of convergent generalised power series.

Examples.

• f ∈ Om, αi ∈ [0, ∞); F (x) = f (xα1

1 , . . . , xαm m ).

The ring of formal generalised power series R x∗. Let x = (x1, . . . , xm). F (x) =

α cαxα such that α ∈ [0, ∞)m, cα ∈ R and

Supp (F) := {α : cα = 0}⊆ S1 × . . . × Sm, where Si ⊆ [0, ∞) well ordered. (F has finitely many minimal monomials) F convergent if ∃r ∈ (0, ∞)m

α |cα|r |α| < ∞ (sup of all finite subsums).

F induces a C0 function (the sum of the series) on [0, r)m (analytic on (0, r)m). R {x∗} =

r R {x∗}r is the ring of convergent generalised power series.

Examples.

• f ∈ Om, αi ∈ [0, ∞); F (x) = f (xα1

1 , . . . , xαm m ). Supp(F) ⊆ α1N × . . . × αmN.

The ring of formal generalised power series R x∗. Let x = (x1, . . . , xm). F (x) =

α cαxα such that α ∈ [0, ∞)m, cα ∈ R and

Supp (F) := {α : cα = 0}⊆ S1 × . . . × Sm, where Si ⊆ [0, ∞) well ordered. (F has finitely many minimal monomials) F convergent if ∃r ∈ (0, ∞)m

α |cα|r |α| < ∞ (sup of all finite subsums).

F induces a C0 function (the sum of the series) on [0, r)m (analytic on (0, r)m). R {x∗} =

r R {x∗}r is the ring of convergent generalised power series.

Examples.

• f ∈ Om, αi ∈ [0, ∞); F (x) = f (xα1

1 , . . . , xαm m ). Supp(F) ⊆ α1N × . . . × αmN.

• F (x) = ζ (− log x)=

n xlog n (Riemann’s ζ).

The ring of formal generalised power series R x∗. Let x = (x1, . . . , xm). F (x) =

α cαxα such that α ∈ [0, ∞)m, cα ∈ R and

Supp (F) := {α : cα = 0}⊆ S1 × . . . × Sm, where Si ⊆ [0, ∞) well ordered. (F has finitely many minimal monomials) F convergent if ∃r ∈ (0, ∞)m

α |cα|r |α| < ∞ (sup of all finite subsums).

F induces a C0 function (the sum of the series) on [0, r)m (analytic on (0, r)m). R {x∗} =

r R {x∗}r is the ring of convergent generalised power series.

Examples.

• f ∈ Om, αi ∈ [0, ∞); F (x) = f (xα1

1 , . . . , xαm m ). Supp(F) ⊆ α1N × . . . × αmN.

• F (x) = ζ (− log x)=

n xlog n (Riemann’s ζ). Supp(F) = {log n}n∈N ր +∞.

The ring of formal generalised power series R x∗. Let x = (x1, . . . , xm). F (x) =

α cαxα such that α ∈ [0, ∞)m, cα ∈ R and

Supp (F) := {α : cα = 0}⊆ S1 × . . . × Sm, where Si ⊆ [0, ∞) well ordered. (F has finitely many minimal monomials) F convergent if ∃r ∈ (0, ∞)m

α |cα|r |α| < ∞ (sup of all finite subsums).

F induces a C0 function (the sum of the series) on [0, r)m (analytic on (0, r)m). R {x∗} =

r R {x∗}r is the ring of convergent generalised power series.

Examples.

• f ∈ Om, αi ∈ [0, ∞); F (x) = f (xα1

1 , . . . , xαm m ). Supp(F) ⊆ α1N × . . . × αmN.

• F (x) = ζ (− log x)=

n xlog n (Riemann’s ζ). Supp(F) = {log n}n∈N ր +∞.

• F (x) =

∞

• n,i=0

1 2i x 2+n− 1

2i

The ring of formal generalised power series R x∗. Let x = (x1, . . . , xm). F (x) =

α cαxα such that α ∈ [0, ∞)m, cα ∈ R and

Supp (F) := {α : cα = 0}⊆ S1 × . . . × Sm, where Si ⊆ [0, ∞) well ordered. (F has finitely many minimal monomials) F convergent if ∃r ∈ (0, ∞)m

α |cα|r |α| < ∞ (sup of all finite subsums).

F induces a C0 function (the sum of the series) on [0, r)m (analytic on (0, r)m). R {x∗} =

r R {x∗}r is the ring of convergent generalised power series.

Examples.

• f ∈ Om, αi ∈ [0, ∞); F (x) = f (xα1

1 , . . . , xαm m ). Supp(F) ⊆ α1N × . . . × αmN.

• F (x) = ζ (− log x)=

n xlog n (Riemann’s ζ). Supp(F) = {log n}n∈N ր +∞.

• F (x) =

∞

• n,i=0

1 2i x 2+n− 1

2i , solution of (1 − x) F (x) = x + 1

2x

• 1 − √x
• F

√x

• .

The ring of formal generalised power series R x∗. Let x = (x1, . . . , xm). F (x) =

α cαxα such that α ∈ [0, ∞)m, cα ∈ R and

Supp (F) := {α : cα = 0}⊆ S1 × . . . × Sm, where Si ⊆ [0, ∞) well ordered. (F has finitely many minimal monomials) F convergent if ∃r ∈ (0, ∞)m

α |cα|r |α| < ∞ (sup of all finite subsums).

F induces a C0 function (the sum of the series) on [0, r)m (analytic on (0, r)m). R {x∗} =

r R {x∗}r is the ring of convergent generalised power series.

Examples.

• f ∈ Om, αi ∈ [0, ∞); F (x) = f (xα1

1 , . . . , xαm m ). Supp(F) ⊆ α1N × . . . × αmN.

• F (x) = ζ (− log x)=

n xlog n (Riemann’s ζ). Supp(F) = {log n}n∈N ր +∞.

• F (x) =

∞

• n,i=0

1 2i x 2+n− 1

2i , solution of (1 − x) F (x) = x + 1

2x

• 1 − √x
• F

√x

• .

1 2 3 4 5 6

The ring of formal generalised power series R x∗. Let x = (x1, . . . , xm). F (x) =

α cαxα such that α ∈ [0, ∞)m, cα ∈ R and

Supp (F) := {α : cα = 0}⊆ S1 × . . . × Sm, where Si ⊆ [0, ∞) well ordered. (F has finitely many minimal monomials) F convergent if ∃r ∈ (0, ∞)m

α |cα|r |α| < ∞ (sup of all finite subsums).

F induces a C0 function (the sum of the series) on [0, r)m (analytic on (0, r)m). R {x∗} =

r R {x∗}r is the ring of convergent generalised power series.

Examples.

• f ∈ Om, αi ∈ [0, ∞); F (x) = f (xα1

1 , . . . , xαm m ). Supp(F) ⊆ α1N × . . . × αmN.

• F (x) = ζ (− log x)=

n xlog n (Riemann’s ζ). Supp(F) = {log n}n∈N ր +∞.

• F (x) =

∞

• n,i=0

1 2i x 2+n− 1

2i , solution of (1 − x) F (x) = x + 1

2x

• 1 − √x
• F

√x

• .

1 2 3 4 5 6

Theorem (vdDries-Speissegger, ’98). R {x∗} generates a polynomially bounded o-minimal expansion Ran∗ of Ran.

Our main result

Our main result

• Definition. A =
• m∈N

R {x∗

1 , . . . , x∗ m} ∪

• x → 1

x

Our main result

• Definition. A =
• m∈N

R {x∗

1 , . . . , x∗ m} ∪

• x → 1

x

• An A-cell is a cell such that the defining functions are A-terms.

Our main result

• Definition. A =
• m∈N

R {x∗

1 , . . . , x∗ m} ∪

• x → 1

x

• An A-cell is a cell such that the defining functions are A-terms.
• THEOREM. x = (x1, . . . , xm) , r ∈ (0, ∞) , f ∈ R {x∗, y ∗}r with f (0, 0) = 0.

Our main result

• Definition. A =
• m∈N

R {x∗

1 , . . . , x∗ m} ∪

• x → 1

x

• An A-cell is a cell such that the defining functions are A-terms.
• THEOREM. x = (x1, . . . , xm) , r ∈ (0, ∞) , f ∈ R {x∗, y ∗}r with f (0, 0) = 0.

Then, ∃ W ⊆ Rm+1 nbd of 0 and ∃ an A-cell decomposition of W ∩ [0, r)m+1

Our main result

• Definition. A =
• m∈N

R {x∗

1 , . . . , x∗ m} ∪

• x → 1

x

• An A-cell is a cell such that the defining functions are A-terms.
• THEOREM. x = (x1, . . . , xm) , r ∈ (0, ∞) , f ∈ R {x∗, y ∗}r with f (0, 0) = 0.

Then, ∃ W ⊆ Rm+1 nbd of 0 and ∃ an A-cell decomposition of W ∩ [0, r)m+1 which is compatible with

• (x, y) ∈ [0, r)m+1 : f (x, y) = 0
• .

Our main result

• Definition. A =
• m∈N

R {x∗

1 , . . . , x∗ m} ∪

• x → 1

x

• An A-cell is a cell such that the defining functions are A-terms.
• THEOREM. x = (x1, . . . , xm) , r ∈ (0, ∞) , f ∈ R {x∗, y ∗}r with f (0, 0) = 0.

Then, ∃ W ⊆ Rm+1 nbd of 0 and ∃ an A-cell decomposition of W ∩ [0, r)m+1 which is compatible with

• (x, y) ∈ [0, r)m+1 : f (x, y) = 0
• .

x y f = 0 f = 0

W

f < 0 f > 0 f < 0 f < 0

Our main result

• Definition. A =
• m∈N

R {x∗

1 , . . . , x∗ m} ∪

• x → 1

x

• An A-cell is a cell such that the defining functions are A-terms.
• THEOREM. x = (x1, . . . , xm) , r ∈ (0, ∞) , f ∈ R {x∗, y ∗}r with f (0, 0) = 0.

Then, ∃ W ⊆ Rm+1 nbd of 0 and ∃ an A-cell decomposition of W ∩ [0, r)m+1 which is compatible with

• (x, y) ∈ [0, r)m+1 : f (x, y) = 0
• .

x y f = 0 f = 0

W

f < 0 f > 0 f < 0 f < 0

In particular, the solutions of f = 0 are of the form y = ϕ (x), where ϕ : C → R is an A-term and C ⊆ Rm is an A-cell.

Strategy of proof

Strategy of proof

Find a finite family F of vertical admissible transformations ρ : [0, ε)m+1 − → [0, r)m+1 (X, Y ) − → (x, y) = (ρ0 (X, Y ) , ρ1 (X, Y )) such that:

Strategy of proof

Find a finite family F of vertical admissible transformations ρ : [0, ε)m+1 − → [0, r)m+1 (X, Y ) − → (x, y) = (ρ0 (X, Y ) , ρ1 (X, Y )) such that: 1) f ◦ ρ ∈ R {X ∗, Y ∗}ε;

Strategy of proof

Find a finite family F of vertical admissible transformations ρ : [0, ε)m+1 − → [0, r)m+1 (X, Y ) − → (x, y) = (ρ0 (X, Y ) , ρ1 (X, Y )) such that: 1) f ◦ ρ ∈ R {X ∗, Y ∗}ε; 2) ∃ W ⊆ Rm+1 nbd of 0 such that W ∩ [0, r)m+1 =

ρ∈F ρ

• [0, ε)m+1

;

Strategy of proof

Find a finite family F of vertical admissible transformations ρ : [0, ε)m+1 − → [0, r)m+1 (X, Y ) − → (x, y) = (ρ0 (X, Y ) , ρ1 (X, Y )) such that: 1) f ◦ ρ ∈ R {X ∗, Y ∗}ε; 2) ∃ W ⊆ Rm+1 nbd of 0 such that W ∩ [0, r)m+1 =

ρ∈F ρ

• [0, ε)m+1

; 3) f ◦ ρ = X αY βU (X, Y ) for some (α, β) ∈ [0, ∞)m+1, U ∈ R {X ∗, Y ∗}× (monomialised form)

Strategy of proof

Find a finite family F of vertical admissible transformations ρ : [0, ε)m+1 − → [0, r)m+1 (X, Y ) − → (x, y) = (ρ0 (X, Y ) , ρ1 (X, Y )) such that: 1) f ◦ ρ ∈ R {X ∗, Y ∗}ε; 2) ∃ W ⊆ Rm+1 nbd of 0 such that W ∩ [0, r)m+1 =

ρ∈F ρ

• [0, ε)m+1

; 3) f ◦ ρ = X αY βU (X, Y ) for some (α, β) ∈ [0, ∞)m+1, U ∈ R {X ∗, Y ∗}× (monomialised form), so f ◦ ρ = 0 has only trivial solutions;

Strategy of proof

Find a finite family F of vertical admissible transformations ρ : [0, ε)m+1 − → [0, r)m+1 (X, Y ) − → (x, y) = (ρ0 (X, Y ) , ρ1 (X, Y )) such that: 1) f ◦ ρ ∈ R {X ∗, Y ∗}ε; 2) ∃ W ⊆ Rm+1 nbd of 0 such that W ∩ [0, r)m+1 =

ρ∈F ρ

• [0, ε)m+1

; 3) f ◦ ρ = X αY βU (X, Y ) for some (α, β) ∈ [0, ∞)m+1, U ∈ R {X ∗, Y ∗}× (monomialised form), so f ◦ ρ = 0 has only trivial solutions; 4) ρ respects y:

Strategy of proof

Find a finite family F of vertical admissible transformations ρ : [0, ε)m+1 − → [0, r)m+1 (X, Y ) − → (x, y) = (ρ0 (X, Y ) , ρ1 (X, Y )) such that: 1) f ◦ ρ ∈ R {X ∗, Y ∗}ε; 2) ∃ W ⊆ Rm+1 nbd of 0 such that W ∩ [0, r)m+1 =

ρ∈F ρ

• [0, ε)m+1

; 3) f ◦ ρ = X αY βU (X, Y ) for some (α, β) ∈ [0, ∞)m+1, U ∈ R {X ∗, Y ∗}× (monomialised form), so f ◦ ρ = 0 has only trivial solutions; 4) ρ respects y:

• ρ0 does not depend on Y , so ρ0 : [0, ε)m ∋ X → x ∈ [0, r)m;

Strategy of proof

Find a finite family F of vertical admissible transformations ρ : [0, ε)m+1 − → [0, r)m+1 (X, Y ) − → (x, y) = (ρ0 (X, Y ) , ρ1 (X, Y )) such that: 1) f ◦ ρ ∈ R {X ∗, Y ∗}ε; 2) ∃ W ⊆ Rm+1 nbd of 0 such that W ∩ [0, r)m+1 =

ρ∈F ρ

• [0, ε)m+1

; 3) f ◦ ρ = X αY βU (X, Y ) for some (α, β) ∈ [0, ∞)m+1, U ∈ R {X ∗, Y ∗}× (monomialised form), so f ◦ ρ = 0 has only trivial solutions; 4) ρ respects y:

• ρ0 does not depend on Y , so ρ0 : [0, ε)m ∋ X → x ∈ [0, r)m;
• ρ0 is bijective outside a small set and the components of ρ−1

are A-terms;

Strategy of proof

Find a finite family F of vertical admissible transformations ρ : [0, ε)m+1 − → [0, r)m+1 (X, Y ) − → (x, y) = (ρ0 (X, Y ) , ρ1 (X, Y )) such that: 1) f ◦ ρ ∈ R {X ∗, Y ∗}ε; 2) ∃ W ⊆ Rm+1 nbd of 0 such that W ∩ [0, r)m+1 =

ρ∈F ρ

• [0, ε)m+1

; 3) f ◦ ρ = X αY βU (X, Y ) for some (α, β) ∈ [0, ∞)m+1, U ∈ R {X ∗, Y ∗}× (monomialised form), so f ◦ ρ = 0 has only trivial solutions; 4) ρ respects y:

• ρ0 does not depend on Y , so ρ0 : [0, ε)m ∋ X → x ∈ [0, r)m;
• ρ0 is bijective outside a small set and the components of ρ−1

are A-terms;

• ρ1 (X, ·) : Y → y is monotonic for almost all X.

Strategy of proof

Find a finite family F of vertical admissible transformations ρ : [0, ε)m+1 − → [0, r)m+1 (X, Y ) − → (x, y) = (ρ0 (X, Y ) , ρ1 (X, Y )) such that: 1) f ◦ ρ ∈ R {X ∗, Y ∗}ε; 2) ∃ W ⊆ Rm+1 nbd of 0 such that W ∩ [0, r)m+1 =

ρ∈F ρ

• [0, ε)m+1

; 3) f ◦ ρ = X αY βU (X, Y ) for some (α, β) ∈ [0, ∞)m+1, U ∈ R {X ∗, Y ∗}× (monomialised form), so f ◦ ρ = 0 has only trivial solutions; 4) ρ respects y:

• ρ0 does not depend on Y , so ρ0 : [0, ε)m ∋ X → x ∈ [0, r)m;
• ρ0 is bijective outside a small set and the components of ρ−1

are A-terms;

• ρ1 (X, ·) : Y → y is monotonic for almost all X.
• Remark. The existence of a family F satisfying 1,2,3 is well known (see

[Rol.-Sanz-Vill.; Rol.-S.’13], inspired by [vdDr.-Speis.’98; Rol.-Speis.-Wil.’03]).

Strategy of proof

Find a finite family F of vertical admissible transformations ρ : [0, ε)m+1 − → [0, r)m+1 (X, Y ) − → (x, y) = (ρ0 (X, Y ) , ρ1 (X, Y )) such that: 1) f ◦ ρ ∈ R {X ∗, Y ∗}ε; 2) ∃ W ⊆ Rm+1 nbd of 0 such that W ∩ [0, r)m+1 =

ρ∈F ρ

• [0, ε)m+1

; 3) f ◦ ρ = X αY βU (X, Y ) for some (α, β) ∈ [0, ∞)m+1, U ∈ R {X ∗, Y ∗}× (monomialised form), so f ◦ ρ = 0 has only trivial solutions; 4) ρ respects y:

• ρ0 does not depend on Y , so ρ0 : [0, ε)m ∋ X → x ∈ [0, r)m;
• ρ0 is bijective outside a small set and the components of ρ−1

are A-terms;

• ρ1 (X, ·) : Y → y is monotonic for almost all X.
• Remark. The existence of a family F satisfying 1,2,3 is well known (see

[Rol.-Sanz-Vill.; Rol.-S.’13], inspired by [vdDr.-Speis.’98; Rol.-Speis.-Wil.’03]). The monomialising tools (admissible transformations) are essentially blow-ups with real exponents and translations by elements of R {x∗}.

Strategy of proof

Find a finite family F of vertical admissible transformations ρ : [0, ε)m+1 − → [0, r)m+1 (X, Y ) − → (x, y) = (ρ0 (X, Y ) , ρ1 (X, Y )) such that: 1) f ◦ ρ ∈ R {X ∗, Y ∗}ε; 2) ∃ W ⊆ Rm+1 nbd of 0 such that W ∩ [0, r)m+1 =

ρ∈F ρ

• [0, ε)m+1

; 3) f ◦ ρ = X αY βU (X, Y ) for some (α, β) ∈ [0, ∞)m+1, U ∈ R {X ∗, Y ∗}× (monomialised form), so f ◦ ρ = 0 has only trivial solutions; 4) ρ respects y:

• ρ0 does not depend on Y , so ρ0 : [0, ε)m ∋ X → x ∈ [0, r)m;
• ρ0 is bijective outside a small set and the components of ρ−1

are A-terms;

• ρ1 (X, ·) : Y → y is monotonic for almost all X.
• Remark. The existence of a family F satisfying 1,2,3 is well known (see

[Rol.-Sanz-Vill.; Rol.-S.’13], inspired by [vdDr.-Speis.’98; Rol.-Speis.-Wil.’03]). The monomialising tools (admissible transformations) are essentially blow-ups with real exponents and translations by elements of R {x∗}. The novelty here lies in 4, which allows to solve f = 0 (verticality).

Why is this enough?

Recall: W ∩ [0, r)m+1 =

ρ∈F ρ

• [0, ε)m+1

and f ◦ ρ = X αY βU (X, Y ); x = ρ0 (X) ; y = ρ1 (X, Y ) .

Why is this enough?

Recall: W ∩ [0, r)m+1 =

ρ∈F ρ

• [0, ε)m+1

and f ◦ ρ = X αY βU (X, Y ); x = ρ0 (X) ; y = ρ1 (X, Y ) . Partition [0, ε)m+1 into finitely many subquadrants Q of dim ≤ m + 1

Why is this enough?

Recall: W ∩ [0, r)m+1 =

ρ∈F ρ

• [0, ε)m+1

and f ◦ ρ = X αY βU (X, Y ); x = ρ0 (X) ; y = ρ1 (X, Y ) . Partition [0, ε)m+1 into finitely many subquadrants Q of dim ≤ m + 1 s.t. f ◦ ρ has constant sign on Q.

Why is this enough?

Recall: W ∩ [0, r)m+1 =

ρ∈F ρ

• [0, ε)m+1

and f ◦ ρ = X αY βU (X, Y ); x = ρ0 (X) ; y = ρ1 (X, Y ) . Partition [0, ε)m+1 into finitely many subquadrants Q of dim ≤ m + 1 s.t. f ◦ ρ has constant sign on Q. Then so does f on ρ (Q).

Why is this enough?

Recall: W ∩ [0, r)m+1 =

ρ∈F ρ

• [0, ε)m+1

and f ◦ ρ = X αY βU (X, Y ); x = ρ0 (X) ; y = ρ1 (X, Y ) . Partition [0, ε)m+1 into finitely many subquadrants Q of dim ≤ m + 1 s.t. f ◦ ρ has constant sign on Q. Then so does f on ρ (Q). So, it is enough to show that ρ (Q) is a finite disjoint union of A-cells.

Why is this enough?

Recall: W ∩ [0, r)m+1 =

ρ∈F ρ

• [0, ε)m+1

and f ◦ ρ = X αY βU (X, Y ); x = ρ0 (X) ; y = ρ1 (X, Y ) . Partition [0, ε)m+1 into finitely many subquadrants Q of dim ≤ m + 1 s.t. f ◦ ρ has constant sign on Q. Then so does f on ρ (Q). So, it is enough to show that ρ (Q) is a finite disjoint union of A-cells. By verticality, ρ0 is invertible and ρ1 (X, ·) is monotonic, outside a small-dimensional set

Why is this enough?

Recall: W ∩ [0, r)m+1 =

ρ∈F ρ

• [0, ε)m+1

and f ◦ ρ = X αY βU (X, Y ); x = ρ0 (X) ; y = ρ1 (X, Y ) . Partition [0, ε)m+1 into finitely many subquadrants Q of dim ≤ m + 1 s.t. f ◦ ρ has constant sign on Q. Then so does f on ρ (Q). So, it is enough to show that ρ (Q) is a finite disjoint union of A-cells. By verticality, ρ0 is invertible and ρ1 (X, ·) is monotonic, outside a small-dimensional set (wlog, an A-cell, by induction on the dimension).

Why is this enough?

Recall: W ∩ [0, r)m+1 =

ρ∈F ρ

• [0, ε)m+1

and f ◦ ρ = X αY βU (X, Y ); x = ρ0 (X) ; y = ρ1 (X, Y ) . Partition [0, ε)m+1 into finitely many subquadrants Q of dim ≤ m + 1 s.t. f ◦ ρ has constant sign on Q. Then so does f on ρ (Q). So, it is enough to show that ρ (Q) is a finite disjoint union of A-cells. By verticality, ρ0 is invertible and ρ1 (X, ·) is monotonic, outside a small-dimensional set (wlog, an A-cell, by induction on the dimension). Hence, it is enough to prove: A ⊆ Rm+1 A-cell and ρ ↾ A as above ⇒ ρ (A) is a fin. disj. union of A-cells.

Why is this enough?

Recall: W ∩ [0, r)m+1 =

ρ∈F ρ

• [0, ε)m+1

and f ◦ ρ = X αY βU (X, Y ); x = ρ0 (X) ; y = ρ1 (X, Y ) . Partition [0, ε)m+1 into finitely many subquadrants Q of dim ≤ m + 1 s.t. f ◦ ρ has constant sign on Q. Then so does f on ρ (Q). So, it is enough to show that ρ (Q) is a finite disjoint union of A-cells. By verticality, ρ0 is invertible and ρ1 (X, ·) is monotonic, outside a small-dimensional set (wlog, an A-cell, by induction on the dimension). Hence, it is enough to prove: A ⊆ Rm+1 A-cell and ρ ↾ A as above ⇒ ρ (A) is a fin. disj. union of A-cells. Wlog, A = {(X, Y ) : X ∈ C, Y ∗ t (X)}, with C ⊆ Rm A-cell, ∗ ∈ {=, <} and t : C → R A-term.

Why is this enough?

Recall: W ∩ [0, r)m+1 =

ρ∈F ρ

• [0, ε)m+1

and f ◦ ρ = X αY βU (X, Y ); x = ρ0 (X) ; y = ρ1 (X, Y ) . Partition [0, ε)m+1 into finitely many subquadrants Q of dim ≤ m + 1 s.t. f ◦ ρ has constant sign on Q. Then so does f on ρ (Q). So, it is enough to show that ρ (Q) is a finite disjoint union of A-cells. By verticality, ρ0 is invertible and ρ1 (X, ·) is monotonic, outside a small-dimensional set (wlog, an A-cell, by induction on the dimension). Hence, it is enough to prove: A ⊆ Rm+1 A-cell and ρ ↾ A as above ⇒ ρ (A) is a fin. disj. union of A-cells. Wlog, A = {(X, Y ) : X ∈ C, Y ∗ t (X)}, with C ⊆ Rm A-cell, ∗ ∈ {=, <} and t : C → R A-term. Then, ρ (A) =

• (x, y) : x ∈ ρ0 (C) , y ∗′ ρ1
• ρ−1

(x) , t

• ρ−1

(x)

• .

Why is this enough?

Recall: W ∩ [0, r)m+1 =

ρ∈F ρ

• [0, ε)m+1

and f ◦ ρ = X αY βU (X, Y ); x = ρ0 (X) ; y = ρ1 (X, Y ) . Partition [0, ε)m+1 into finitely many subquadrants Q of dim ≤ m + 1 s.t. f ◦ ρ has constant sign on Q. Then so does f on ρ (Q). So, it is enough to show that ρ (Q) is a finite disjoint union of A-cells. By verticality, ρ0 is invertible and ρ1 (X, ·) is monotonic, outside a small-dimensional set (wlog, an A-cell, by induction on the dimension). Hence, it is enough to prove: A ⊆ Rm+1 A-cell and ρ ↾ A as above ⇒ ρ (A) is a fin. disj. union of A-cells. Wlog, A = {(X, Y ) : X ∈ C, Y ∗ t (X)}, with C ⊆ Rm A-cell, ∗ ∈ {=, <} and t : C → R A-term. Then, ρ (A) =

• (x, y) : x ∈ ρ0 (C) , y ∗′ ρ1
• ρ−1

(x) , t

• ρ−1

(x)

• .

By induction on the dimension, ρ0 (C) is a fin. disj. union of A-cells.

Why is this enough?

Recall: W ∩ [0, r)m+1 =

ρ∈F ρ

• [0, ε)m+1

and f ◦ ρ = X αY βU (X, Y ); x = ρ0 (X) ; y = ρ1 (X, Y ) . Partition [0, ε)m+1 into finitely many subquadrants Q of dim ≤ m + 1 s.t. f ◦ ρ has constant sign on Q. Then so does f on ρ (Q). So, it is enough to show that ρ (Q) is a finite disjoint union of A-cells. By verticality, ρ0 is invertible and ρ1 (X, ·) is monotonic, outside a small-dimensional set (wlog, an A-cell, by induction on the dimension). Hence, it is enough to prove: A ⊆ Rm+1 A-cell and ρ ↾ A as above ⇒ ρ (A) is a fin. disj. union of A-cells. Wlog, A = {(X, Y ) : X ∈ C, Y ∗ t (X)}, with C ⊆ Rm A-cell, ∗ ∈ {=, <} and t : C → R A-term. Then, ρ (A) =

• (x, y) : x ∈ ρ0 (C) , y ∗′ ρ1
• ρ−1

(x) , t

• ρ−1

(x)

• .

By induction on the dimension, ρ0 (C) is a fin. disj. union of A-cells. By verticality, ρ1

• ρ−1

(x) , t

• ρ−1

(x)

• is an A-term.

Why is this enough?

Recall: W ∩ [0, r)m+1 =

ρ∈F ρ

• [0, ε)m+1

and f ◦ ρ = X αY βU (X, Y ); x = ρ0 (X) ; y = ρ1 (X, Y ) . Partition [0, ε)m+1 into finitely many subquadrants Q of dim ≤ m + 1 s.t. f ◦ ρ has constant sign on Q. Then so does f on ρ (Q). So, it is enough to show that ρ (Q) is a finite disjoint union of A-cells. By verticality, ρ0 is invertible and ρ1 (X, ·) is monotonic, outside a small-dimensional set (wlog, an A-cell, by induction on the dimension). Hence, it is enough to prove: A ⊆ Rm+1 A-cell and ρ ↾ A as above ⇒ ρ (A) is a fin. disj. union of A-cells. Wlog, A = {(X, Y ) : X ∈ C, Y ∗ t (X)}, with C ⊆ Rm A-cell, ∗ ∈ {=, <} and t : C → R A-term. Then, ρ (A) =

• (x, y) : x ∈ ρ0 (C) , y ∗′ ρ1
• ρ−1

(x) , t

• ρ−1

(x)

• .

By induction on the dimension, ρ0 (C) is a fin. disj. union of A-cells. By verticality, ρ1

• ρ−1

(x) , t

• ρ−1

(x)

• is an A-term.

Examples of vertical blow-ups (m=2)

Examples of vertical blow-ups (m=2)

Fix δ ∈ (0, ∞).        π0 : (x1, x2, y) →

• x1, xδ

1 x2, y

• (chart at 0)

πλ : (x1, x2, y) →

• x1, xδ

1 (λ + x2) , y

• (λ ∈ R\ {0})

(regular charts) π∞ : (x1, x2, y) →

• x1x1/δ

2

, x2, y

• (chart at ∞)

Examples of vertical blow-ups (m=2)

Fix δ ∈ (0, ∞).        π0 : (x1, x2, y) →

• x1, xδ

1 x2, y

• (chart at 0)

πλ : (x1, x2, y) →

• x1, xδ

1 (λ + x2) , y

• (λ ∈ R\ {0})

(regular charts) π∞ : (x1, x2, y) →

• x1x1/δ

2

, x2, y

• (chart at ∞)
• If f =
• α,β,γ

cαβγxα

1 xβ 2 y γ and λ ∈ R ∪ {∞} , then f ◦ πλ ∈ R {x∗ 1 , x∗ 2 , y ∗} :

Examples of vertical blow-ups (m=2)

Fix δ ∈ (0, ∞).        π0 : (x1, x2, y) →

• x1, xδ

1 x2, y

• (chart at 0)

πλ : (x1, x2, y) →

• x1, xδ

1 (λ + x2) , y

• (λ ∈ R\ {0})

(regular charts) π∞ : (x1, x2, y) →

• x1x1/δ

2

, x2, y

• (chart at ∞)
• If f =
• α,β,γ

cαβγxα

1 xβ 2 y γ and λ ∈ R ∪ {∞} , then f ◦ πλ ∈ R {x∗ 1 , x∗ 2 , y ∗} :

f

• x1, xδ

1 (λ + x2) , y

• =

α,β,γ cαβγxα+δβ 1

(λ + x2)β y γ

Examples of vertical blow-ups (m=2)

Fix δ ∈ (0, ∞).        π0 : (x1, x2, y) →

• x1, xδ

1 x2, y

• (chart at 0)

πλ : (x1, x2, y) →

• x1, xδ

1 (λ + x2) , y

• (λ ∈ R\ {0})

(regular charts) π∞ : (x1, x2, y) →

• x1x1/δ

2

, x2, y

• (chart at ∞)
• If f =
• α,β,γ

cαβγxα

1 xβ 2 y γ and λ ∈ R ∪ {∞} , then f ◦ πλ ∈ R {x∗ 1 , x∗ 2 , y ∗} :

f

• x1, xδ

1 (λ + x2) , y

• =

α,β,γ cαβγxα+δβ 1

(λ + x2)β y γ (λ + x2)β =

n∈N

β

n

• λβ−nxn

2

Examples of vertical blow-ups (m=2)

Fix δ ∈ (0, ∞).        π0 : (x1, x2, y) →

• x1, xδ

1 x2, y

• (chart at 0)

πλ : (x1, x2, y) →

• x1, xδ

1 (λ + x2) , y

• (λ ∈ R\ {0})

(regular charts) π∞ : (x1, x2, y) →

• x1x1/δ

2

, x2, y

• (chart at ∞)
• If f =
• α,β,γ

cαβγxα

1 xβ 2 y γ and λ ∈ R ∪ {∞} , then f ◦ πλ ∈ R {x∗ 1 , x∗ 2 , y ∗} :

f

• x1, xδ

1 (λ + x2) , y

• =

α,β,γ cαβγxα+δβ 1

(λ + x2)β y γ (λ + x2)β =

n∈N

β

n

• λβ−nxn

2

(x2 becomes an analytic variable)

Examples of vertical blow-ups (m=2)

Fix δ ∈ (0, ∞).        π0 : (x1, x2, y) →

• x1, xδ

1 x2, y

• (chart at 0)

πλ : (x1, x2, y) →

• x1, xδ

1 (λ + x2) , y

• (λ ∈ R\ {0})

(regular charts) π∞ : (x1, x2, y) →

• x1x1/δ

2

, x2, y

• (chart at ∞)
• If f =
• α,β,γ

cαβγxα

1 xβ 2 y γ and λ ∈ R ∪ {∞} , then f ◦ πλ ∈ R {x∗ 1 , x∗ 2 , y ∗} :

f

• x1, xδ

1 (λ + x2) , y

• =

α,β,γ cαβγxα+δβ 1

(λ + x2)β y γ (λ + x2)β =

n∈N

β

n

• λβ−nxn

2

(x2 becomes an analytic variable)

• 

      π0 [0, ε)3 =

• (x1, x2, y) : 0 ≤ x1 < ε, 0 ≤ x2 < εxδ

1

• πλ

[0, ε)3 =

• (x1, x2, y) : 0 ≤ x1 < ε, λxδ

1 ≤ x2 < (λ + ε) xδ 1

• π∞

[0, ε)3 =

• (x1, x2, y) : 0 ≤ x1 < εx1/δ

2

, 0 ≤ x2 < ε

Examples of vertical blow-ups (m=2)

Fix δ ∈ (0, ∞).        π0 : (x1, x2, y) →

• x1, xδ

1 x2, y

• (chart at 0)

πλ : (x1, x2, y) →

• x1, xδ

1 (λ + x2) , y

• (λ ∈ R\ {0})

(regular charts) π∞ : (x1, x2, y) →

• x1x1/δ

2

, x2, y

• (chart at ∞)
• If f =
• α,β,γ

cαβγxα

1 xβ 2 y γ and λ ∈ R ∪ {∞} , then f ◦ πλ ∈ R {x∗ 1 , x∗ 2 , y ∗} :

f

• x1, xδ

1 (λ + x2) , y

• =

α,β,γ cαβγxα+δβ 1

(λ + x2)β y γ (λ + x2)β =

n∈N

β

n

• λβ−nxn

2

(x2 becomes an analytic variable)

• 

      π0 [0, ε)3 =

• (x1, x2, y) : 0 ≤ x1 < ε, 0 ≤ x2 < εxδ

1

• πλ

[0, ε)3 =

• (x1, x2, y) : 0 ≤ x1 < ε, λxδ

1 ≤ x2 < (λ + ε) xδ 1

• π∞

[0, ε)3 =

• (x1, x2, y) : 0 ≤ x1 < εx1/δ

2

, 0 ≤ x2 < ε

• , so

[0, r)3 ∩ W =

• λ∈R∪{∞}

πλ [0, ε)3

Examples of vertical blow-ups (m=2)

Fix δ ∈ (0, ∞).        π0 : (x1, x2, y) →

• x1, xδ

1 x2, y

• (chart at 0)

πλ : (x1, x2, y) →

• x1, xδ

1 (λ + x2) , y

• (λ ∈ R\ {0})

(regular charts) π∞ : (x1, x2, y) →

• x1x1/δ

2

, x2, y

• (chart at ∞)
• If f =
• α,β,γ

cαβγxα

1 xβ 2 y γ and λ ∈ R ∪ {∞} , then f ◦ πλ ∈ R {x∗ 1 , x∗ 2 , y ∗} :

f

• x1, xδ

1 (λ + x2) , y

• =

α,β,γ cαβγxα+δβ 1

(λ + x2)β y γ (λ + x2)β =

n∈N

β

n

• λβ−nxn

2

(x2 becomes an analytic variable)

• 

      π0 [0, ε)3 =

• (x1, x2, y) : 0 ≤ x1 < ε, 0 ≤ x2 < εxδ

1

• πλ

[0, ε)3 =

• (x1, x2, y) : 0 ≤ x1 < ε, λxδ

1 ≤ x2 < (λ + ε) xδ 1

• π∞

[0, ε)3 =

• (x1, x2, y) : 0 ≤ x1 < εx1/δ

2

, 0 ≤ x2 < ε

• , so

[0, r)3 ∩ W =

• λ∈R∪{∞}

πλ [0, ε)3 (need only finitely many λ, by compactness)

Examples of vertical blow-ups (m=2)

Fix δ ∈ (0, ∞).        π0 : (x1, x2, y) →

• x1, xδ

1 x2, y

• (chart at 0)

πλ : (x1, x2, y) →

• x1, xδ

1 (λ + x2) , y

• (λ ∈ R\ {0})

(regular charts) π∞ : (x1, x2, y) →

• x1x1/δ

2

, x2, y

• (chart at ∞)
• If f =
• α,β,γ

cαβγxα

1 xβ 2 y γ and λ ∈ R ∪ {∞} , then f ◦ πλ ∈ R {x∗ 1 , x∗ 2 , y ∗} :

f

• x1, xδ

1 (λ + x2) , y

• =

α,β,γ cαβγxα+δβ 1

(λ + x2)β y γ (λ + x2)β =

n∈N

β

n

• λβ−nxn

2

(x2 becomes an analytic variable)

• 

      π0 [0, ε)3 =

• (x1, x2, y) : 0 ≤ x1 < ε, 0 ≤ x2 < εxδ

1

• πλ

[0, ε)3 =

• (x1, x2, y) : 0 ≤ x1 < ε, λxδ

1 ≤ x2 < (λ + ε) xδ 1

• π∞

[0, ε)3 =

• (x1, x2, y) : 0 ≤ x1 < εx1/δ

2

, 0 ≤ x2 < ε

• , so

[0, r)3 ∩ W =

• λ∈R∪{∞}

πλ [0, ε)3 (need only finitely many λ, by compactness)

• πλ =
• πλ

0 , πλ 1

• respects y, for λ ∈ R ∪ {∞}:

πλ

1 =id (so, monotonic),

Examples of vertical blow-ups (m=2)

Fix δ ∈ (0, ∞).        π0 : (x1, x2, y) →

• x1, xδ

1 x2, y

• (chart at 0)

πλ : (x1, x2, y) →

• x1, xδ

1 (λ + x2) , y

• (λ ∈ R\ {0})

(regular charts) π∞ : (x1, x2, y) →

• x1x1/δ

2

, x2, y

• (chart at ∞)
• If f =
• α,β,γ

cαβγxα

1 xβ 2 y γ and λ ∈ R ∪ {∞} , then f ◦ πλ ∈ R {x∗ 1 , x∗ 2 , y ∗} :

f

• x1, xδ

1 (λ + x2) , y

• =

α,β,γ cαβγxα+δβ 1

(λ + x2)β y γ (λ + x2)β =

n∈N

β

n

• λβ−nxn

2

(x2 becomes an analytic variable)

• 

      π0 [0, ε)3 =

• (x1, x2, y) : 0 ≤ x1 < ε, 0 ≤ x2 < εxδ

1

• πλ

[0, ε)3 =

• (x1, x2, y) : 0 ≤ x1 < ε, λxδ

1 ≤ x2 < (λ + ε) xδ 1

• π∞

[0, ε)3 =

• (x1, x2, y) : 0 ≤ x1 < εx1/δ

2

, 0 ≤ x2 < ε

• , so

[0, r)3 ∩ W =

• λ∈R∪{∞}

πλ [0, ε)3 (need only finitely many λ, by compactness)

• πλ =
• πλ

0 , πλ 1

• respects y, for λ ∈ R ∪ {∞}:

πλ

1 =id (so, monotonic),

πλ

0 does not depend on y and is bijective outside {x1 = 0} , {x2 = 0}

Examples of vertical blow-ups (m=2)

Fix δ ∈ (0, ∞).        π0 : (x1, x2, y) →

• x1, xδ

1 x2, y

• (chart at 0)

πλ : (x1, x2, y) →

• x1, xδ

1 (λ + x2) , y

• (λ ∈ R\ {0})

(regular charts) π∞ : (x1, x2, y) →

• x1x1/δ

2

, x2, y

• (chart at ∞)
• If f =
• α,β,γ

cαβγxα

1 xβ 2 y γ and λ ∈ R ∪ {∞} , then f ◦ πλ ∈ R {x∗ 1 , x∗ 2 , y ∗} :

f

• x1, xδ

1 (λ + x2) , y

• =

α,β,γ cαβγxα+δβ 1

(λ + x2)β y γ (λ + x2)β =

n∈N

β

n

• λβ−nxn

2

(x2 becomes an analytic variable)

• 

      π0 [0, ε)3 =

• (x1, x2, y) : 0 ≤ x1 < ε, 0 ≤ x2 < εxδ

1

• πλ

[0, ε)3 =

• (x1, x2, y) : 0 ≤ x1 < ε, λxδ

1 ≤ x2 < (λ + ε) xδ 1

• π∞

[0, ε)3 =

• (x1, x2, y) : 0 ≤ x1 < εx1/δ

2

, 0 ≤ x2 < ε

• , so

[0, r)3 ∩ W =

• λ∈R∪{∞}

πλ [0, ε)3 (need only finitely many λ, by compactness)

• πλ =
• πλ

0 , πλ 1

• respects y, for λ ∈ R ∪ {∞}:

πλ

1 =id (so, monotonic),

πλ

0 does not depend on y and is bijective outside {x1 = 0} , {x2 = 0}

• πλ

−1:(x1, x2) →

• x1, x2x−δ

1

• ;
• x1, x2x−δ

1

− λ

• ;
• x1x−1/δ

2

, x2

• A-terms

Blow-ups in action

• Example. f (x1, x2, y) = xα

1 − xβ 2 y γ (1 + x1y)

Blow-ups in action

• Example. f (x1, x2, y) = xα

1 − xβ 2 y γ (1 + x1y)

• π blow-up of (x1, x2) with exponent δ = α

β

Blow-ups in action

• Example. f (x1, x2, y) = xα

1 − xβ 2 y γ (1 + x1y)

• π blow-up of (x1, x2) with exponent δ = α

β

         f ◦ π0 = f

• x1, xα/β

1

x2, y

• f ◦ πλ = f
• x1, xα/β

1

(λ + x2) , y

• f ◦ π∞ = f
• x1x1/δ

2

, x2, y

Blow-ups in action

• Example. f (x1, x2, y) = xα

1 − xβ 2 y γ (1 + x1y)

• π blow-up of (x1, x2) with exponent δ = α

β

         f ◦ π0 = f

• x1, xα/β

1

x2, y

• = xα

1

• 1 − xβ

2 y γ (1 + x1y)

• f ◦ πλ = f
• x1, xα/β

1

(λ + x2) , y

• f ◦ π∞ = f
• x1x1/δ

2

, x2, y

Blow-ups in action

• Example. f (x1, x2, y) = xα

1 − xβ 2 y γ (1 + x1y)

• π blow-up of (x1, x2) with exponent δ = α

β

         f ◦ π0 = f

• x1, xα/β

1

x2, y

• = xα

1

• 1 − xβ

2 y γ (1 + x1y)

• = xα

1 · unit

f ◦ πλ = f

• x1, xα/β

1

(λ + x2) , y

• f ◦ π∞ = f
• x1x1/δ

2

, x2, y

Blow-ups in action

• Example. f (x1, x2, y) = xα

1 − xβ 2 y γ (1 + x1y)

• π blow-up of (x1, x2) with exponent δ = α

β

         f ◦ π0 = f

• x1, xα/β

1

x2, y

• = xα

1

• 1 − xβ

2 y γ (1 + x1y)

• = xα

1 · unit

f ◦ πλ = f

• x1, xα/β

1

(λ + x2) , y

• = xα

1

• 1 − y γ (1 + x1y) (λ + x2)β

f ◦ π∞ = f

• x1x1/δ

2

, x2, y

Blow-ups in action

• Example. f (x1, x2, y) = xα

1 − xβ 2 y γ (1 + x1y)

• π blow-up of (x1, x2) with exponent δ = α

β

         f ◦ π0 = f

• x1, xα/β

1

x2, y

• = xα

1

• 1 − xβ

2 y γ (1 + x1y)

• = xα

1 · unit

f ◦ πλ = f

• x1, xα/β

1

(λ + x2) , y

• = xα

1

• 1 − y γ (1 + x1y) (λ + x2)β

= xα

1 · unit

f ◦ π∞ = f

• x1x1/δ

2

, x2, y

Blow-ups in action

• Example. f (x1, x2, y) = xα

1 − xβ 2 y γ (1 + x1y)

• π blow-up of (x1, x2) with exponent δ = α

β

         f ◦ π0 = f

• x1, xα/β

1

x2, y

• = xα

1

• 1 − xβ

2 y γ (1 + x1y)

• = xα

1 · unit

f ◦ πλ = f

• x1, xα/β

1

(λ + x2) , y

• = xα

1

• 1 − y γ (1 + x1y) (λ + x2)β

= xα

1 · unit

f ◦ π∞ = f

• x1x1/δ

2

, x2, y

• = xβ

2

• xα

1 − y γ

1 + x1xβ/α

2

y

Blow-ups in action

• Example. f (x1, x2, y) = xα

1 − xβ 2 y γ (1 + x1y)

• π blow-up of (x1, x2) with exponent δ = α

β

         f ◦ π0 = f

• x1, xα/β

1

x2, y

• = xα

1

• 1 − xβ

2 y γ (1 + x1y)

• = xα

1 · unit

f ◦ πλ = f

• x1, xα/β

1

(λ + x2) , y

• = xα

1

• 1 − y γ (1 + x1y) (λ + x2)β

= xα

1 · unit

f ◦ π∞ = f

• x1x1/δ

2

, x2, y

• = xβ

2

• xα

1 − y γ

1 + x1xβ/α

2

y

• Now we look at g (x1, x2, y) = xα

1 − y γ

1 + x1xβ/α

2

y

Blow-ups in action

• Example. f (x1, x2, y) = xα

1 − xβ 2 y γ (1 + x1y)

• π blow-up of (x1, x2) with exponent δ = α

β

         f ◦ π0 = f

• x1, xα/β

1

x2, y

• = xα

1

• 1 − xβ

2 y γ (1 + x1y)

• = xα

1 · unit

f ◦ πλ = f

• x1, xα/β

1

(λ + x2) , y

• = xα

1

• 1 − y γ (1 + x1y) (λ + x2)β

= xα

1 · unit

f ◦ π∞ = f

• x1x1/δ

2

, x2, y

• = xβ

2

• xα

1 − y γ

1 + x1xβ/α

2

y

• Now we look at g (x1, x2, y) = xα

1 − y γ

1 + x1xβ/α

2

y

• ˜

π = blow-up of (x1, y) with exponent α

γ

Blow-ups in action

• Example. f (x1, x2, y) = xα

1 − xβ 2 y γ (1 + x1y)

• π blow-up of (x1, x2) with exponent δ = α

β

         f ◦ π0 = f

• x1, xα/β

1

x2, y

• = xα

1

• 1 − xβ

2 y γ (1 + x1y)

• = xα

1 · unit

f ◦ πλ = f

• x1, xα/β

1

(λ + x2) , y

• = xα

1

• 1 − y γ (1 + x1y) (λ + x2)β

= xα

1 · unit

f ◦ π∞ = f

• x1x1/δ

2

, x2, y

• = xβ

2

• xα

1 − y γ

1 + x1xβ/α

2

y

• Now we look at g (x1, x2, y) = xα

1 − y γ

1 + x1xβ/α

2

y

• ˜

π = blow-up of (x1, y) with exponent α

γ

a) chart at 0 ˜ π0 : (x1, x2, y) →

• x1, x2, xα/γ

1

y

Blow-ups in action

• Example. f (x1, x2, y) = xα

1 − xβ 2 y γ (1 + x1y)

• π blow-up of (x1, x2) with exponent δ = α

β

         f ◦ π0 = f

• x1, xα/β

1

x2, y

• = xα

1

• 1 − xβ

2 y γ (1 + x1y)

• = xα

1 · unit

f ◦ πλ = f

• x1, xα/β

1

(λ + x2) , y

• = xα

1

• 1 − y γ (1 + x1y) (λ + x2)β

= xα

1 · unit

f ◦ π∞ = f

• x1x1/δ

2

, x2, y

• = xβ

2

• xα

1 − y γ

1 + x1xβ/α

2

y

• Now we look at g (x1, x2, y) = xα

1 − y γ

1 + x1xβ/α

2

y

• ˜

π = blow-up of (x1, y) with exponent α

γ

a) chart at 0 ˜ π0 : (x1, x2, y) →

• x1, x2, xα/γ

1

y

• g ◦ ˜

π0 = xα

1 (1 − y γ (1 + η0 (x1, x2, y))), with η0 (0, 0, 0) = 0.

Blow-ups in action

• Example. f (x1, x2, y) = xα

1 − xβ 2 y γ (1 + x1y)

• π blow-up of (x1, x2) with exponent δ = α

β

         f ◦ π0 = f

• x1, xα/β

1

x2, y

• = xα

1

• 1 − xβ

2 y γ (1 + x1y)

• = xα

1 · unit

f ◦ πλ = f

• x1, xα/β

1

(λ + x2) , y

• = xα

1

• 1 − y γ (1 + x1y) (λ + x2)β

= xα

1 · unit

f ◦ π∞ = f

• x1x1/δ

2

, x2, y

• = xβ

2

• xα

1 − y γ

1 + x1xβ/α

2

y

• Now we look at g (x1, x2, y) = xα

1 − y γ

1 + x1xβ/α

2

y

• ˜

π = blow-up of (x1, y) with exponent α

γ

a) chart at 0 ˜ π0 : (x1, x2, y) →

• x1, x2, xα/γ

1

y

• g ◦ ˜

π0 = xα

1 (1 − y γ (1 + η0 (x1, x2, y))), with η0 (0, 0, 0) = 0. (trivial solution)

Blow-ups in action

• Example. f (x1, x2, y) = xα

1 − xβ 2 y γ (1 + x1y)

• π blow-up of (x1, x2) with exponent δ = α

β

         f ◦ π0 = f

• x1, xα/β

1

x2, y

• = xα

1

• 1 − xβ

2 y γ (1 + x1y)

• = xα

1 · unit

f ◦ πλ = f

• x1, xα/β

1

(λ + x2) , y

• = xα

1

• 1 − y γ (1 + x1y) (λ + x2)β

= xα

1 · unit

f ◦ π∞ = f

• x1x1/δ

2

, x2, y

• = xβ

2

• xα

1 − y γ

1 + x1xβ/α

2

y

• Now we look at g (x1, x2, y) = xα

1 − y γ

1 + x1xβ/α

2

y

• ˜

π = blow-up of (x1, y) with exponent α

γ

a) chart at 0 ˜ π0 : (x1, x2, y) →

• x1, x2, xα/γ

1

y

• g ◦ ˜

π0 = xα

1 (1 − y γ (1 + η0 (x1, x2, y))), with η0 (0, 0, 0) = 0. (trivial solution)

b) regular charts for λ ∈ R \ {0} ˜ πλ : (x1, x2, y) →

• x1, x2, xα/γ

1

(λ + y)

Blow-ups in action

• Example. f (x1, x2, y) = xα

1 − xβ 2 y γ (1 + x1y)

• π blow-up of (x1, x2) with exponent δ = α

β

         f ◦ π0 = f

• x1, xα/β

1

x2, y

• = xα

1

• 1 − xβ

2 y γ (1 + x1y)

• = xα

1 · unit

f ◦ πλ = f

• x1, xα/β

1

(λ + x2) , y

• = xα

1

• 1 − y γ (1 + x1y) (λ + x2)β

= xα

1 · unit

f ◦ π∞ = f

• x1x1/δ

2

, x2, y

• = xβ

2

• xα

1 − y γ

1 + x1xβ/α

2

y

• Now we look at g (x1, x2, y) = xα

1 − y γ

1 + x1xβ/α

2

y

• ˜

π = blow-up of (x1, y) with exponent α

γ

a) chart at 0 ˜ π0 : (x1, x2, y) →

• x1, x2, xα/γ

1

y

• g ◦ ˜

π0 = xα

1 (1 − y γ (1 + η0 (x1, x2, y))), with η0 (0, 0, 0) = 0. (trivial solution)

b) regular charts for λ ∈ R \ {0} ˜ πλ : (x1, x2, y) →

• x1, x2, xα/γ

1

(λ + y)

• g ◦ ˜

πλ = xα

1 (1 − (λ + y)γ (1 + η0 (x1, x2, y)))

Blow-ups in action

• Example. f (x1, x2, y) = xα

1 − xβ 2 y γ (1 + x1y)

• π blow-up of (x1, x2) with exponent δ = α

β

         f ◦ π0 = f

• x1, xα/β

1

x2, y

• = xα

1

• 1 − xβ

2 y γ (1 + x1y)

• = xα

1 · unit

f ◦ πλ = f

• x1, xα/β

1

(λ + x2) , y

• = xα

1

• 1 − y γ (1 + x1y) (λ + x2)β

= xα

1 · unit

f ◦ π∞ = f

• x1x1/δ

2

, x2, y

• = xβ

2

• xα

1 − y γ

1 + x1xβ/α

2

y

• Now we look at g (x1, x2, y) = xα

1 − y γ

1 + x1xβ/α

2

y

• ˜

π = blow-up of (x1, y) with exponent α

γ

a) chart at 0 ˜ π0 : (x1, x2, y) →

• x1, x2, xα/γ

1

y

• g ◦ ˜

π0 = xα

1 (1 − y γ (1 + η0 (x1, x2, y))), with η0 (0, 0, 0) = 0. (trivial solution)

b) regular charts for λ ∈ R \ {0} ˜ πλ : (x1, x2, y) →

• x1, x2, xα/γ

1

(λ + y)

• g ◦ ˜

πλ = xα

1 (1 − (λ + y)γ (1 + η0 (x1, x2, y)))

= xα

1 (1 − λγ + η (x1, x2, y)), with η (0, 0, 0) = 0

Blow-ups in action

• Example. f (x1, x2, y) = xα

1 − xβ 2 y γ (1 + x1y)

• π blow-up of (x1, x2) with exponent δ = α

β

         f ◦ π0 = f

• x1, xα/β

1

x2, y

• = xα

1

• 1 − xβ

2 y γ (1 + x1y)

• = xα

1 · unit

f ◦ πλ = f

• x1, xα/β

1

(λ + x2) , y

• = xα

1

• 1 − y γ (1 + x1y) (λ + x2)β

= xα

1 · unit

f ◦ π∞ = f

• x1x1/δ

2

, x2, y

• = xβ

2

• xα

1 − y γ

1 + x1xβ/α

2

y

• Now we look at g (x1, x2, y) = xα

1 − y γ

1 + x1xβ/α

2

y

• ˜

π = blow-up of (x1, y) with exponent α

γ

a) chart at 0 ˜ π0 : (x1, x2, y) →

• x1, x2, xα/γ

1

y

• g ◦ ˜

π0 = xα

1 (1 − y γ (1 + η0 (x1, x2, y))), with η0 (0, 0, 0) = 0. (trivial solution)

b) regular charts for λ ∈ R \ {0} ˜ πλ : (x1, x2, y) →

• x1, x2, xα/γ

1

(λ + y)

• g ◦ ˜

πλ = xα

1 (1 − (λ + y)γ (1 + η0 (x1, x2, y)))

= xα

1 (1 − λγ + η (x1, x2, y)), with η (0, 0, 0) = 0

(y analytic)

Blow-ups in action

• Example. f (x1, x2, y) = xα

1 − xβ 2 y γ (1 + x1y)

• π blow-up of (x1, x2) with exponent δ = α

β

         f ◦ π0 = f

• x1, xα/β

1

x2, y

• = xα

1

• 1 − xβ

2 y γ (1 + x1y)

• = xα

1 · unit

f ◦ πλ = f

• x1, xα/β

1

(λ + x2) , y

• = xα

1

• 1 − y γ (1 + x1y) (λ + x2)β

= xα

1 · unit

f ◦ π∞ = f

• x1x1/δ

2

, x2, y

• = xβ

2

• xα

1 − y γ

1 + x1xβ/α

2

y

• Now we look at g (x1, x2, y) = xα

1 − y γ

1 + x1xβ/α

2

y

• ˜

π = blow-up of (x1, y) with exponent α

γ

a) chart at 0 ˜ π0 : (x1, x2, y) →

• x1, x2, xα/γ

1

y

• g ◦ ˜

π0 = xα

1 (1 − y γ (1 + η0 (x1, x2, y))), with η0 (0, 0, 0) = 0. (trivial solution)

b) regular charts for λ ∈ R \ {0} ˜ πλ : (x1, x2, y) →

• x1, x2, xα/γ

1

(λ + y)

• g ◦ ˜

πλ = xα

1 (1 − (λ + y)γ (1 + η0 (x1, x2, y)))

= xα

1 (1 − λγ + η (x1, x2, y)), with η (0, 0, 0) = 0

(y analytic) c) chart at ∞ ˜ π∞ : (x1, x2, y) →

• x1y γ/α, x2, y

Blow-ups in action

• Example. f (x1, x2, y) = xα

1 − xβ 2 y γ (1 + x1y)

• π blow-up of (x1, x2) with exponent δ = α

β

         f ◦ π0 = f

• x1, xα/β

1

x2, y

• = xα

1

• 1 − xβ

2 y γ (1 + x1y)

• = xα

1 · unit

f ◦ πλ = f

• x1, xα/β

1

(λ + x2) , y

• = xα

1

• 1 − y γ (1 + x1y) (λ + x2)β

= xα

1 · unit

f ◦ π∞ = f

• x1x1/δ

2

, x2, y

• = xβ

2

• xα

1 − y γ

1 + x1xβ/α

2

y

• Now we look at g (x1, x2, y) = xα

1 − y γ

1 + x1xβ/α

2

y

• ˜

π = blow-up of (x1, y) with exponent α

γ

a) chart at 0 ˜ π0 : (x1, x2, y) →

• x1, x2, xα/γ

1

y

• g ◦ ˜

π0 = xα

1 (1 − y γ (1 + η0 (x1, x2, y))), with η0 (0, 0, 0) = 0. (trivial solution)

b) regular charts for λ ∈ R \ {0} ˜ πλ : (x1, x2, y) →

• x1, x2, xα/γ

1

(λ + y)

• g ◦ ˜

πλ = xα

1 (1 − (λ + y)γ (1 + η0 (x1, x2, y)))

= xα

1 (1 − λγ + η (x1, x2, y)), with η (0, 0, 0) = 0

(y analytic) c) chart at ∞ ˜ π∞ : (x1, x2, y) →

• x1y γ/α, x2, y
• ! NOT VERTICAL !

Blow-ups in action

• Example. f (x1, x2, y) = xα

1 − xβ 2 y γ (1 + x1y)

• π blow-up of (x1, x2) with exponent δ = α

β

         f ◦ π0 = f

• x1, xα/β

1

x2, y

• = xα

1

• 1 − xβ

2 y γ (1 + x1y)

• = xα

1 · unit

f ◦ πλ = f

• x1, xα/β

1

(λ + x2) , y

• = xα

1

• 1 − y γ (1 + x1y) (λ + x2)β

= xα

1 · unit

f ◦ π∞ = f

• x1x1/δ

2

, x2, y

• = xβ

2

• xα

1 − y γ

1 + x1xβ/α

2

y

• Now we look at g (x1, x2, y) = xα

1 − y γ

1 + x1xβ/α

2

y

• ˜

π = blow-up of (x1, y) with exponent α

γ

a) chart at 0 ˜ π0 : (x1, x2, y) →

• x1, x2, xα/γ

1

y

• g ◦ ˜

π0 = xα

1 (1 − y γ (1 + η0 (x1, x2, y))), with η0 (0, 0, 0) = 0. (trivial solution)

b) regular charts for λ ∈ R \ {0} ˜ πλ : (x1, x2, y) →

• x1, x2, xα/γ

1

(λ + y)

• g ◦ ˜

πλ = xα

1 (1 − (λ + y)γ (1 + η0 (x1, x2, y)))

= xα

1 (1 − λγ + η (x1, x2, y)), with η (0, 0, 0) = 0

(y analytic) c) chart at ∞ ˜ π∞ : (x1, x2, y) →

• x1y γ/α, x2, y
• ! NOT VERTICAL !

However, y γ > > xα

1 on ˜

π∞ [0, ε)3 , so g cannot vanish there.

Blow-ups in action

• Example. f (x1, x2, y) = xα

1 − xβ 2 y γ (1 + x1y)

• π blow-up of (x1, x2) with exponent δ = α

β

         f ◦ π0 = f

• x1, xα/β

1

x2, y

• = xα

1

• 1 − xβ

2 y γ (1 + x1y)

• = xα

1 · unit

f ◦ πλ = f

• x1, xα/β

1

(λ + x2) , y

• = xα

1

• 1 − y γ (1 + x1y) (λ + x2)β

= xα

1 · unit

f ◦ π∞ = f

• x1x1/δ

2

, x2, y

• = xβ

2

• xα

1 − y γ

1 + x1xβ/α

2

y

• Now we look at g (x1, x2, y) = xα

1 − y γ

1 + x1xβ/α

2

y

• ˜

π = blow-up of (x1, y) with exponent α

γ

a) chart at 0 ˜ π0 : (x1, x2, y) →

• x1, x2, xα/γ

1

y

• g ◦ ˜

π0 = xα

1 (1 − y γ (1 + η0 (x1, x2, y))), with η0 (0, 0, 0) = 0. (trivial solution)

b) regular charts for λ ∈ R \ {0} ˜ πλ : (x1, x2, y) →

• x1, x2, xα/γ

1

(λ + y)

• g ◦ ˜

πλ = xα

1 (1 − (λ + y)γ (1 + η0 (x1, x2, y)))

= xα

1 (1 − λγ + η (x1, x2, y)), with η (0, 0, 0) = 0

(y analytic) c) chart at ∞ ˜ π∞ : (x1, x2, y) →

• x1y γ/α, x2, y
• ! NOT VERTICAL !

However, y γ > > xα

1 on ˜

π∞ [0, ε)3 , so g cannot vanish there. (trivial solution)

General strategy. f (x1, x2, y) = d

i=1 xαi 1 xβi 2 y γi Ui (x1, x2, y)

Ui units.

General strategy. f (x1, x2, y) = d

i=1 xαi 1 xβi 2 y γi Ui (x1, x2, y)

Ui units. Monomialisation algorithm: the set of minimal monomials

• xαi

1 xβi 2 y γi

d

i=1

determines the choice of pairs of variables and exponents in the blow-ups.

General strategy. f (x1, x2, y) = d

i=1 xαi 1 xβi 2 y γi Ui (x1, x2, y)

Ui units. Monomialisation algorithm: the set of minimal monomials

• xαi

1 xβi 2 y γi

d

i=1

determines the choice of pairs of variables and exponents in the blow-ups. ———

General strategy. f (x1, x2, y) = d

i=1 xαi 1 xβi 2 y γi Ui (x1, x2, y)

Ui units. Monomialisation algorithm: the set of minimal monomials

• xαi

1 xβi 2 y γi

d

i=1

determines the choice of pairs of variables and exponents in the blow-ups. ——— Problem: solving systems of equations (joint work with J.-P. Rolin). Given f , g ∈ R {x∗

1 , x∗ 2 , y ∗ 1 , y ∗ 2 }, find the solutions of

• f (x1, x2, y1, y2) = 0

g (x1, x2, y1, y2) = 0.

General strategy. f (x1, x2, y) = d

i=1 xαi 1 xβi 2 y γi Ui (x1, x2, y)

Ui units. Monomialisation algorithm: the set of minimal monomials

• xαi

1 xβi 2 y γi

d

i=1

determines the choice of pairs of variables and exponents in the blow-ups. ——— Problem: solving systems of equations (joint work with J.-P. Rolin). Given f , g ∈ R {x∗

1 , x∗ 2 , y ∗ 1 , y ∗ 2 }, find the solutions of

• f (x1, x2, y1, y2) = 0

g (x1, x2, y1, y2) = 0. Possible strategy. Find a solution y2 = t (x1, x2, y1) (t A-term) of f = 0 and replace in g.

General strategy. f (x1, x2, y) = d

i=1 xαi 1 xβi 2 y γi Ui (x1, x2, y)

Ui units. Monomialisation algorithm: the set of minimal monomials

• xαi

1 xβi 2 y γi

d

i=1

determines the choice of pairs of variables and exponents in the blow-ups. ——— Problem: solving systems of equations (joint work with J.-P. Rolin). Given f , g ∈ R {x∗

1 , x∗ 2 , y ∗ 1 , y ∗ 2 }, find the solutions of

• f (x1, x2, y1, y2) = 0

g (x1, x2, y1, y2) = 0. Possible strategy. Find a solution y2 = t (x1, x2, y1) (t A-term) of f = 0 and replace in g. Then solve g (x1, x2, y1, t (x1, x2, y1)) = 0 wrto y1.

General strategy. f (x1, x2, y) = d

i=1 xαi 1 xβi 2 y γi Ui (x1, x2, y)

Ui units. Monomialisation algorithm: the set of minimal monomials

• xαi

1 xβi 2 y γi

d

i=1

determines the choice of pairs of variables and exponents in the blow-ups. ——— Problem: solving systems of equations (joint work with J.-P. Rolin). Given f , g ∈ R {x∗

1 , x∗ 2 , y ∗ 1 , y ∗ 2 }, find the solutions of

• f (x1, x2, y1, y2) = 0

g (x1, x2, y1, y2) = 0. Possible strategy. Find a solution y2 = t (x1, x2, y1) (t A-term) of f = 0 and replace in g. Then solve g (x1, x2, y1, t (x1, x2, y1)) = 0 wrto y1. Complicated!

General strategy. f (x1, x2, y) = d

i=1 xαi 1 xβi 2 y γi Ui (x1, x2, y)

Ui units. Monomialisation algorithm: the set of minimal monomials

• xαi

1 xβi 2 y γi

d

i=1

determines the choice of pairs of variables and exponents in the blow-ups. ——— Problem: solving systems of equations (joint work with J.-P. Rolin). Given f , g ∈ R {x∗

1 , x∗ 2 , y ∗ 1 , y ∗ 2 }, find the solutions of

• f (x1, x2, y1, y2) = 0

g (x1, x2, y1, y2) = 0. Possible strategy. Find a solution y2 = t (x1, x2, y1) (t A-term) of f = 0 and replace in g. Then solve g (x1, x2, y1, t (x1, x2, y1)) = 0 wrto y1. Complicated!

• Example. g (x1, x2, y1, t (x1, x2, y1)) = h
• x1, x2, x1

y1

• , where h ∈ R {x∗

1 , x∗ 2 , y ∗ 1 }.

General strategy. f (x1, x2, y) = d

i=1 xαi 1 xβi 2 y γi Ui (x1, x2, y)

Ui units. Monomialisation algorithm: the set of minimal monomials

• xαi

1 xβi 2 y γi

d

i=1

determines the choice of pairs of variables and exponents in the blow-ups. ——— Problem: solving systems of equations (joint work with J.-P. Rolin). Given f , g ∈ R {x∗

1 , x∗ 2 , y ∗ 1 , y ∗ 2 }, find the solutions of

• f (x1, x2, y1, y2) = 0

g (x1, x2, y1, y2) = 0. Possible strategy. Find a solution y2 = t (x1, x2, y1) (t A-term) of f = 0 and replace in g. Then solve g (x1, x2, y1, t (x1, x2, y1)) = 0 wrto y1. Complicated!

• Example. g (x1, x2, y1, t (x1, x2, y1)) = h
• x1, x2, x1

y1

• , where h ∈ R {x∗

1 , x∗ 2 , y ∗ 1 }.

Not a generalised power series, no monomialisation algorithm.

General strategy. f (x1, x2, y) = d

i=1 xαi 1 xβi 2 y γi Ui (x1, x2, y)

Ui units. Monomialisation algorithm: the set of minimal monomials

• xαi

1 xβi 2 y γi

d

i=1

determines the choice of pairs of variables and exponents in the blow-ups. ——— Problem: solving systems of equations (joint work with J.-P. Rolin). Given f , g ∈ R {x∗

1 , x∗ 2 , y ∗ 1 , y ∗ 2 }, find the solutions of

• f (x1, x2, y1, y2) = 0

g (x1, x2, y1, y2) = 0. Possible strategy. Find a solution y2 = t (x1, x2, y1) (t A-term) of f = 0 and replace in g. Then solve g (x1, x2, y1, t (x1, x2, y1)) = 0 wrto y1. Complicated!

• Example. g (x1, x2, y1, t (x1, x2, y1)) = h
• x1, x2, x1

y1

• , where h ∈ R {x∗

1 , x∗ 2 , y ∗ 1 }.

Not a generalised power series, no monomialisation algorithm.

• Idea. Terms like x/y come from charts at ∞ (not vertical), so avoid them!

General strategy. f (x1, x2, y) = d

i=1 xαi 1 xβi 2 y γi Ui (x1, x2, y)

Ui units. Monomialisation algorithm: the set of minimal monomials

• xαi

1 xβi 2 y γi

d

i=1

determines the choice of pairs of variables and exponents in the blow-ups. ——— Problem: solving systems of equations (joint work with J.-P. Rolin). Given f , g ∈ R {x∗

1 , x∗ 2 , y ∗ 1 , y ∗ 2 }, find the solutions of

• f (x1, x2, y1, y2) = 0

g (x1, x2, y1, y2) = 0. Possible strategy. Find a solution y2 = t (x1, x2, y1) (t A-term) of f = 0 and replace in g. Then solve g (x1, x2, y1, t (x1, x2, y1)) = 0 wrto y1. Complicated!

• Example. g (x1, x2, y1, t (x1, x2, y1)) = h
• x1, x2, x1

y1

• , where h ∈ R {x∗

1 , x∗ 2 , y ∗ 1 }.

Not a generalised power series, no monomialisation algorithm.

• Idea. Terms like x/y come from charts at ∞ (not vertical), so avoid them!

Black box: using an o-minimal preparation theorem (vdDries-Speiss.), we prove that,

General strategy. f (x1, x2, y) = d

i=1 xαi 1 xβi 2 y γi Ui (x1, x2, y)

Ui units. Monomialisation algorithm: the set of minimal monomials

• xαi

1 xβi 2 y γi

d

i=1

determines the choice of pairs of variables and exponents in the blow-ups. ——— Problem: solving systems of equations (joint work with J.-P. Rolin). Given f , g ∈ R {x∗

1 , x∗ 2 , y ∗ 1 , y ∗ 2 }, find the solutions of

• f (x1, x2, y1, y2) = 0

g (x1, x2, y1, y2) = 0. Possible strategy. Find a solution y2 = t (x1, x2, y1) (t A-term) of f = 0 and replace in g. Then solve g (x1, x2, y1, t (x1, x2, y1)) = 0 wrto y1. Complicated!

• Example. g (x1, x2, y1, t (x1, x2, y1)) = h
• x1, x2, x1

y1

• , where h ∈ R {x∗

1 , x∗ 2 , y ∗ 1 }.

Not a generalised power series, no monomialisation algorithm.

• Idea. Terms like x/y come from charts at ∞ (not vertical), so avoid them!

Black box: using an o-minimal preparation theorem (vdDries-Speiss.), we prove that, if

• y1 = ϕ1 (x1, x2)

y2 = ϕ2 (x1, x2)is a solution of

• f = 0

g = 0

General strategy. f (x1, x2, y) = d

i=1 xαi 1 xβi 2 y γi Ui (x1, x2, y)

Ui units. Monomialisation algorithm: the set of minimal monomials

• xαi

1 xβi 2 y γi

d

i=1

determines the choice of pairs of variables and exponents in the blow-ups. ——— Problem: solving systems of equations (joint work with J.-P. Rolin). Given f , g ∈ R {x∗

1 , x∗ 2 , y ∗ 1 , y ∗ 2 }, find the solutions of

• f (x1, x2, y1, y2) = 0

g (x1, x2, y1, y2) = 0. Possible strategy. Find a solution y2 = t (x1, x2, y1) (t A-term) of f = 0 and replace in g. Then solve g (x1, x2, y1, t (x1, x2, y1)) = 0 wrto y1. Complicated!

• Example. g (x1, x2, y1, t (x1, x2, y1)) = h
• x1, x2, x1

y1

• , where h ∈ R {x∗

1 , x∗ 2 , y ∗ 1 }.

Not a generalised power series, no monomialisation algorithm.

• Idea. Terms like x/y come from charts at ∞ (not vertical), so avoid them!

Black box: using an o-minimal preparation theorem (vdDries-Speiss.), we prove that, if

• y1 = ϕ1 (x1, x2)

y2 = ϕ2 (x1, x2)is a solution of

• f = 0

g = 0, then, after suitable blow-ups, ∃ αi, βi ∈ [0, ∞) such that 1

2xαi 1 xβi 2

≤ ϕi (x1, x2) ≤ 3

2xαi 1 xβi 2 .

General strategy. f (x1, x2, y) = d

i=1 xαi 1 xβi 2 y γi Ui (x1, x2, y)

Ui units. Monomialisation algorithm: the set of minimal monomials

• xαi

1 xβi 2 y γi

d

i=1

determines the choice of pairs of variables and exponents in the blow-ups. ——— Problem: solving systems of equations (joint work with J.-P. Rolin). Given f , g ∈ R {x∗

1 , x∗ 2 , y ∗ 1 , y ∗ 2 }, find the solutions of

• f (x1, x2, y1, y2) = 0

g (x1, x2, y1, y2) = 0. Possible strategy. Find a solution y2 = t (x1, x2, y1) (t A-term) of f = 0 and replace in g. Then solve g (x1, x2, y1, t (x1, x2, y1)) = 0 wrto y1. Complicated!

• Example. g (x1, x2, y1, t (x1, x2, y1)) = h
• x1, x2, x1

y1

• , where h ∈ R {x∗

1 , x∗ 2 , y ∗ 1 }.

Not a generalised power series, no monomialisation algorithm.

• Idea. Terms like x/y come from charts at ∞ (not vertical), so avoid them!

Black box: using an o-minimal preparation theorem (vdDries-Speiss.), we prove that, if

• y1 = ϕ1 (x1, x2)

y2 = ϕ2 (x1, x2)is a solution of

• f = 0

g = 0, then, after suitable blow-ups, ∃ αi, βi ∈ [0, ∞) such that 1

2xαi 1 xβi 2

≤ ϕi (x1, x2) ≤ 3

2xαi 1 xβi 2 .

We monomialise vertically simultaneously f , g using this piece of information.

General strategy. f (x1, x2, y) = d

i=1 xαi 1 xβi 2 y γi Ui (x1, x2, y)

Ui units. Monomialisation algorithm: the set of minimal monomials

• xαi

1 xβi 2 y γi

d

i=1

determines the choice of pairs of variables and exponents in the blow-ups. ——— Problem: solving systems of equations (joint work with J.-P. Rolin). Given f , g ∈ R {x∗

1 , x∗ 2 , y ∗ 1 , y ∗ 2 }, find the solutions of

• f (x1, x2, y1, y2) = 0

g (x1, x2, y1, y2) = 0. Possible strategy. Find a solution y2 = t (x1, x2, y1) (t A-term) of f = 0 and replace in g. Then solve g (x1, x2, y1, t (x1, x2, y1)) = 0 wrto y1. Complicated!

• Example. g (x1, x2, y1, t (x1, x2, y1)) = h
• x1, x2, x1

y1

• , where h ∈ R {x∗

1 , x∗ 2 , y ∗ 1 }.

Not a generalised power series, no monomialisation algorithm.

• Idea. Terms like x/y come from charts at ∞ (not vertical), so avoid them!

Black box: using an o-minimal preparation theorem (vdDries-Speiss.), we prove that, if

• y1 = ϕ1 (x1, x2)

y2 = ϕ2 (x1, x2)is a solution of

• f = 0

g = 0, then, after suitable blow-ups, ∃ αi, βi ∈ [0, ∞) such that 1

2xαi 1 xβi 2

≤ ϕi (x1, x2) ≤ 3

2xαi 1 xβi 2 .

We monomialise vertically simultaneously f , g using this piece of information. Unfortunately, this strategy is not algorithmic, unlike the previous one.

Other classes to which the main result applies

Other classes to which the main result applies

1) Quasianalytic Denjoy-Carleman classes: 1 ≥ M0 ≥ M1 ≥ . . . sequence of real numbers such that M2

i ≤ i i+1Mi−1Mi+1.

Other classes to which the main result applies

1) Quasianalytic Denjoy-Carleman classes: 1 ≥ M0 ≥ M1 ≥ . . . sequence of real numbers such that M2

i ≤ i i+1Mi−1Mi+1.

Cn (M) = collection of all f ∈ C∞ ([−1, 1]n) such that there exists A > 0 with ∀α ∈ Nn, ∀x ∈ [−1, 1]n ,

• ∂αf

∂xα (x)

• ≤ A|α|+1M|α|

Other classes to which the main result applies

1) Quasianalytic Denjoy-Carleman classes: 1 ≥ M0 ≥ M1 ≥ . . . sequence of real numbers such that M2

i ≤ i i+1Mi−1Mi+1.

Cn (M) = collection of all f ∈ C∞ ([−1, 1]n) such that there exists A > 0 with ∀α ∈ Nn, ∀x ∈ [−1, 1]n ,

• ∂αf

∂xα (x)

• ≤ A|α|+1M|α|

Cn (M) is quasianalytic if and only if ∞

i=0 Mi Mi+1 = ∞

Other classes to which the main result applies

1) Quasianalytic Denjoy-Carleman classes: 1 ≥ M0 ≥ M1 ≥ . . . sequence of real numbers such that M2

i ≤ i i+1Mi−1Mi+1.

Cn (M) = collection of all f ∈ C∞ ([−1, 1]n) such that there exists A > 0 with ∀α ∈ Nn, ∀x ∈ [−1, 1]n ,

• ∂αf

∂xα (x)

• ≤ A|α|+1M|α|

Cn (M) is quasianalytic if and only if ∞

i=0 Mi Mi+1 = ∞ (e.g. Mi = (i log i)i).

Other classes to which the main result applies

1) Quasianalytic Denjoy-Carleman classes: 1 ≥ M0 ≥ M1 ≥ . . . sequence of real numbers such that M2

i ≤ i i+1Mi−1Mi+1.

Cn (M) = collection of all f ∈ C∞ ([−1, 1]n) such that there exists A > 0 with ∀α ∈ Nn, ∀x ∈ [−1, 1]n ,

• ∂αf

∂xα (x)

• ≤ A|α|+1M|α|

Cn (M) is quasianalytic if and only if ∞

i=0 Mi Mi+1 = ∞ (e.g. Mi = (i log i)i).

2) Multisummable series (as in [vdDries-Speissegger, ’00]): A collection of C∞ functions on [0, r]n satisfying a multivariable Gevrey-like growth condition.

Other classes to which the main result applies

1) Quasianalytic Denjoy-Carleman classes: 1 ≥ M0 ≥ M1 ≥ . . . sequence of real numbers such that M2

i ≤ i i+1Mi−1Mi+1.

Cn (M) = collection of all f ∈ C∞ ([−1, 1]n) such that there exists A > 0 with ∀α ∈ Nn, ∀x ∈ [−1, 1]n ,

• ∂αf

∂xα (x)

• ≤ A|α|+1M|α|

Cn (M) is quasianalytic if and only if ∞

i=0 Mi Mi+1 = ∞ (e.g. Mi = (i log i)i).

2) Multisummable series (as in [vdDries-Speissegger, ’00]): A collection of C∞ functions on [0, r]n satisfying a multivariable Gevrey-like growth condition. For example, the function ψ appearing in:

Other classes to which the main result applies

1) Quasianalytic Denjoy-Carleman classes: 1 ≥ M0 ≥ M1 ≥ . . . sequence of real numbers such that M2

i ≤ i i+1Mi−1Mi+1.

Cn (M) = collection of all f ∈ C∞ ([−1, 1]n) such that there exists A > 0 with ∀α ∈ Nn, ∀x ∈ [−1, 1]n ,

• ∂αf

∂xα (x)

• ≤ A|α|+1M|α|

Cn (M) is quasianalytic if and only if ∞

i=0 Mi Mi+1 = ∞ (e.g. Mi = (i log i)i).

2) Multisummable series (as in [vdDries-Speissegger, ’00]): A collection of C∞ functions on [0, r]n satisfying a multivariable Gevrey-like growth condition. For example, the function ψ appearing in: (Binet’s second formula) log Γ (x) =

• x − 1

2

• log x − x + 1

2 log (2π) + ψ

1

x

Other classes to which the main result applies

1) Quasianalytic Denjoy-Carleman classes: 1 ≥ M0 ≥ M1 ≥ . . . sequence of real numbers such that M2

i ≤ i i+1Mi−1Mi+1.

Cn (M) = collection of all f ∈ C∞ ([−1, 1]n) such that there exists A > 0 with ∀α ∈ Nn, ∀x ∈ [−1, 1]n ,

• ∂αf

∂xα (x)

• ≤ A|α|+1M|α|

Cn (M) is quasianalytic if and only if ∞

i=0 Mi Mi+1 = ∞ (e.g. Mi = (i log i)i).

2) Multisummable series (as in [vdDries-Speissegger, ’00]): A collection of C∞ functions on [0, r]n satisfying a multivariable Gevrey-like growth condition. For example, the function ψ appearing in: (Binet’s second formula) log Γ (x) =

• x − 1

2

• log x − x + 1

2 log (2π) + ψ

1

x

• 3) AH-analytic functions (as in [Rolin-Sanz-Schaefke, ’07]):

Let H (x) : [0, ε) → R be a solution of Euler’s differential eq. x2y ′ = y − x.

Other classes to which the main result applies

1) Quasianalytic Denjoy-Carleman classes: 1 ≥ M0 ≥ M1 ≥ . . . sequence of real numbers such that M2

i ≤ i i+1Mi−1Mi+1.

Cn (M) = collection of all f ∈ C∞ ([−1, 1]n) such that there exists A > 0 with ∀α ∈ Nn, ∀x ∈ [−1, 1]n ,

• ∂αf

∂xα (x)

• ≤ A|α|+1M|α|

Cn (M) is quasianalytic if and only if ∞

i=0 Mi Mi+1 = ∞ (e.g. Mi = (i log i)i).

2) Multisummable series (as in [vdDries-Speissegger, ’00]): A collection of C∞ functions on [0, r]n satisfying a multivariable Gevrey-like growth condition. For example, the function ψ appearing in: (Binet’s second formula) log Γ (x) =

• x − 1

2

• log x − x + 1

2 log (2π) + ψ

1

x

• 3) AH-analytic functions (as in [Rolin-Sanz-Schaefke, ’07]):

Let H (x) : [0, ε) → R be a solution of Euler’s differential eq. x2y ′ = y − x. AH = the smallest collection of real C∞ germs containing H and closed under composition, monomial division and implicit functions.

Other classes to which the main result applies

1) Quasianalytic Denjoy-Carleman classes: 1 ≥ M0 ≥ M1 ≥ . . . sequence of real numbers such that M2

i ≤ i i+1Mi−1Mi+1.

Cn (M) = collection of all f ∈ C∞ ([−1, 1]n) such that there exists A > 0 with ∀α ∈ Nn, ∀x ∈ [−1, 1]n ,

• ∂αf

∂xα (x)

• ≤ A|α|+1M|α|

Cn (M) is quasianalytic if and only if ∞

i=0 Mi Mi+1 = ∞ (e.g. Mi = (i log i)i).

2) Multisummable series (as in [vdDries-Speissegger, ’00]): A collection of C∞ functions on [0, r]n satisfying a multivariable Gevrey-like growth condition. For example, the function ψ appearing in: (Binet’s second formula) log Γ (x) =

• x − 1

2

• log x − x + 1

2 log (2π) + ψ

1

x

• 3) AH-analytic functions (as in [Rolin-Sanz-Schaefke, ’07]):

Let H (x) : [0, ε) → R be a solution of Euler’s differential eq. x2y ′ = y − x. AH = the smallest collection of real C∞ germs containing H and closed under composition, monomial division and implicit functions. 4) The class Q (as in [Kaiser-Rolin-Speissegger, ’09]): A collection containing the Dulac transition maps of real analytic planar vector fields in a neighbourhood of hyperbolic non-resonant singular points.