The Mysterious Actual Order of the Pole ˆof the Igusa ......

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  • The Mysterious Actual Order of the Pole ρ of the Igusa Local Zeta

    Function Exploration of a Conjecture

    Joanna Miles [email protected]

    Mount Holyoke College Mathematics REU 2005

    – p.1/14

  • Overview • Our main interest: the poles of the Igusa local zeta function.

    • Pole: a value for which the denominator of a function vanishes.

    • There are many candidate poles, with expected orders.

    These are well understood.

    • Not all candidate poles are actual poles. One special actual

    pole, called ρ, is well understood for certain classes of

    functions.

    • For other classes of functions, very little is known about ρ

    and its order.

    • We studied one of the latter classes.

    – p.2/14

  • Overview • Our main interest: the poles of the Igusa local zeta function.

    • Pole: a value for which the denominator of a function vanishes.

    • There are many candidate poles, with expected orders.

    These are well understood.

    • Not all candidate poles are actual poles. One special actual

    pole, called ρ, is well understood for certain classes of

    functions.

    • For other classes of functions, very little is known about ρ

    and its order.

    • We studied one of the latter classes.

    – p.2/14

  • Overview • Our main interest: the poles of the Igusa local zeta function.

    • Pole: a value for which the denominator of a function vanishes.

    • There are many candidate poles, with expected orders.

    These are well understood.

    • Not all candidate poles are actual poles. One special actual

    pole, called ρ, is well understood for certain classes of

    functions.

    • For other classes of functions, very little is known about ρ

    and its order.

    • We studied one of the latter classes.

    – p.2/14

  • Overview • Our main interest: the poles of the Igusa local zeta function.

    • Pole: a value for which the denominator of a function vanishes.

    • There are many candidate poles, with expected orders.

    These are well understood.

    • Not all candidate poles are actual poles. One special actual

    pole, called ρ, is well understood for certain classes of

    functions.

    • For other classes of functions, very little is known about ρ

    and its order.

    • We studied one of the latter classes.

    – p.2/14

  • Overview • Our main interest: the poles of the Igusa local zeta function.

    • Pole: a value for which the denominator of a function vanishes.

    • There are many candidate poles, with expected orders.

    These are well understood.

    • Not all candidate poles are actual poles. One special actual

    pole, called ρ, is well understood for certain classes of

    functions.

    • For other classes of functions, very little is known about ρ

    and its order.

    • We studied one of the latter classes.

    – p.2/14

  • Overview • Our main interest: the poles of the Igusa local zeta function.

    • Pole: a value for which the denominator of a function vanishes.

    • There are many candidate poles, with expected orders.

    These are well understood.

    • Not all candidate poles are actual poles. One special actual

    pole, called ρ, is well understood for certain classes of

    functions.

    • For other classes of functions, very little is known about ρ

    and its order.

    • We studied one of the latter classes.

    – p.2/14

  • p-adic Numbers What is a p-adic integer?

    • Fix p prime. • If m is a p-adic integer, ∃ a unique p-adic

    expansion

    m = a0 + a1p + a2p 2 + a3p

    3 + . . .

    with 0 ≤ ai ≤ p − 1. • Zp: the set of all p-adic integers.

    – p.3/14

  • The Igusa Local Zeta Function Let f(x) be a polynomial in n variables with integer coefficients.

    Zf(s) =

    Zn p

    |f(x)|sp dx

    where s ∈ C for Re(s) > 0.

    Zf(s) is always a rational function of t = p−s (Igusa,

    1975).

    – p.4/14

  • Structure of the Newton Polygon

    Newton polygon Γ(f) for f(x, y) = x3y2 + x5 − xy3:

    • Support points: (3, 2), (5, 0), (1, 3)

    • Faces: Intersections of Γ(f) with

    supporting hyperplanes. We have 6.

    • Each face has associated

    polynomial fτ .

    • Cones: associated to the faces, form

    a partition of the first quadrant.

    • The cones give us regions of Zp for

    integration.

    – p.5/14

  • Structure of the Newton Polygon

    Newton polygon Γ(f) for f(x, y) = x3y2 + x5 − xy3:

    • Support points: (3, 2), (5, 0), (1, 3)

    • Faces: Intersections of Γ(f) with

    supporting hyperplanes. We have 6.

    • Each face has associated

    polynomial fτ .

    • Cones: associated to the faces, form

    a partition of the first quadrant.

    • The cones give us regions of Zp for

    integration.

    – p.5/14

  • Structure of the Newton Polygon

    Newton polygon Γ(f) for f(x, y) = x3y2 + x5 − xy3:

    • Support points: (3, 2), (5, 0), (1, 3)

    • Faces: Intersections of Γ(f) with

    supporting hyperplanes. We have 6.

    • Each face has associated

    polynomial fτ .

    • Cones: associated to the faces, form

    a partition of the first quadrant.

    • The cones give us regions of Zp for

    integration.

    – p.5/14

  • Structure of the Newton Polygon

    Newton polygon Γ(f) for f(x, y) = x3y2 + x5 − xy3:

    • Support points: (3, 2), (5, 0), (1, 3)

    • Faces: Intersections of Γ(f) with

    supporting hyperplanes. We have 6.

    • Each face has associated

    polynomial fτ .

    • Cones: associated to the faces, form

    a partition of the first quadrant.

    • The cones give us regions of Zp for

    integration.

    – p.5/14

  • Structure of the Newton Polygon

    Newton polygon Γ(f) for f(x, y) = x3y2 + x5 − xy3:

    • Support points: (3, 2), (5, 0), (1, 3)

    • Faces: Intersections of Γ(f) with

    supporting hyperplanes. We have 6.

    • Each face has associated

    polynomial fτ .

    • Cones: associated to the faces, form

    a partition of the first quadrant.

    • The cones give us regions of Zp for

    integration.

    – p.5/14

  • Calculation of Zf(s) Using the Newton Polygon

    Method Let f(x) be non-degenerate: it has no singular points mod p in

    the p-adic units for any fτ (x).

    Zf (s) = ∑

    τ face of Γ(f) LτS∆τ

    where

    Lτ = p −n

    (

    (p − 1)n − p|Nτ |

    (

    ps − 1

    ps+1 − 1

    ))

    S∆τ = ∑

    k∈Nn∩∆τ

    p−σ(k)−m(k)s

    σ(k) = Pn

    i=1 ki

    m(a) = infx∈Γ(f){a · x}

    Nτ contains all x such that fτ (x) ≡ 0 mod p – p.6/14

  • Our Candidate Poles and Expected Orders

    Pole: a value s for which the denominator of Zf (s) vanishes. Candidate poles come from 2 places:

    1. (ps+1 − 1) term in denominator of Lτ

    2. Denominators of S∆τ come from (p σ(ai) − pm(ai)s) where ai

    is a spanning vector of ∆τ .

    The expected order of a candidate pole (other than −1) is the

    maximal order of that pole from all S∆τ .

    Of course, when we add terms, some of these poles disappear. . .

    – p.7/14

  • Our Candidate Poles and Expected Orders

    Pole: a value s for which the denominator of Zf (s) vanishes. Candidate poles come from 2 places:

    1. (ps+1 − 1) term in denominator of Lτ

    2. Denominators of S∆τ come from (p σ(ai) − pm(ai)s) where ai

    is a spanning vector of ∆τ .

    The expected order of a candidate pole (other than −1) is the

    maximal order of that pole from all S∆τ .

    Of course, when we add terms, some of these poles disappear. . .

    – p.7/14

  • Our Candidate Poles and Expected Orders

    Pole: a value s for which the denominator of Zf (s) vanishes. Candidate poles come from 2 places:

    1. (ps+1 − 1) term in denominator of Lτ

    2. Denominators of S∆τ come from (p σ(ai) − pm(ai)s) where ai

    is a spanning vector of ∆τ .

    The expected order of a candidate pole (other than −1) is the

    maximal order of that pole from all S∆τ .

    Of course, when we add terms, some of these poles disappear. . .

    – p.7/14

  • Our Candidate Poles and Expected Orders

    Pole: a value s for which the denominator of Zf (s) vanishes. Candidate poles come from 2 places:

    1. (ps+1 − 1) term in denominator of Lτ

    2. Denominators of S∆τ come from (p σ(ai) − pm(ai)s) where ai

    is a spanning vector of ∆τ .

    The expected order of a candidate pole (other than −1) is the

    maximal order of that pole from all S∆τ .

    Of course, when we add terms, some of these poles disappear. . .

    – p.7/14

  • Our Favorite Candidate Pole: t0 • (t0, . . . , t0): the unique point of

    intersection of the boundary of

    Γ(f) and the line (t, t, . . . , t)