On Products of Delta Distributions and Resultants
We prove an identity in integral geometry, showing that if ₓ and ₓ are two polynomials, ∫dxδ( ₓ) ⊗ δ( ₓ) is proportional to δ( ) where is the resultant of ₓ and ₓ.
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| description | We prove an identity in integral geometry, showing that if ₓ and ₓ are two polynomials, ∫dxδ( ₓ) ⊗ δ( ₓ) is proportional to δ( ) where is the resultant of ₓ and ₓ.
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 16 (2020), 083, 11 pages
On Products of Delta Distributions and Resultants
Michel BAUER †1†2†3†4†5 and Jean-Bernard ZUBER †6†7
†1 Institut de Physique Théorique de Saclay, CEA-Saclay, F-91191 Gif-sur-Yvette, France
E-mail: michel.bauer@ipht.fr
†2 CNRS, UMR 3681, IPhT, F-91191 Gif-sur-Yvette, France
†3 Département de mathématiques et applications, École normale supérieure,
F-75005 Paris, France
†4 CNRS, UMR 8553, DMA, ENS, F-75005 Paris, France
†5 PSL Research University, F-75005 Paris, France
†6 Sorbonne Université, UMR 7589, LPTHE, F-75005, Paris, France
E-mail: jean-bernard.zuber@upmc.fr
†7 CNRS, UMR 7589, LPTHE, F-75005, Paris, France
Received June 16, 2020, in final form August 20, 2020; Published online August 25, 2020
https://doi.org/10.3842/SIGMA.2020.083
Abstract. We prove an identity in integral geometry, showing that if Px and Qx are two
polynomials,
∫
dx δ(Px) ⊗ δ(Qx) is proportional to δ(R) where R is the resultant of Px
and Qx.
Key words: measures and distributions; integral geometry
2020 Mathematics Subject Classification: 46F10; 49Q15
1 Introduction
In their classical text [2, see in particular Chapter 3], Gelfand and Shilov introduce generalized
functions localized on smooth submanifolds of Rn. A most important example is δ(f) with
f : Rn → R a C1 function whose gradient Grad f vanishes nowhere on f = 0.
In this note, we introduce a variant of the original definition of δ(f) and extend it to deal
with certain cases when f and Grad f have common zeros. A salient feature of this extension is
that δ(f), when defined, is no longer always a distribution, but a positive σ-finite measure.
Our aim is to prove the following combinatorial identity:
Proposition 1. Let A, B be disjoint finite sets. For x ∈ R, a, b ∈ R∗, u := (uα)α∈A ∈ RA
and v := (vβ)β∈B ∈ RB, set Px(u) := a
∏
α∈A(x − uα) and Qx(v) := b
∏
α∈B(x − vβ). Let
R(u, v) := a|B|b|A|
∏
α∈A
β∈B
(uα − vβ) be the resultant of Px and Qx seen as polynomials in the
variable x. Set
J(u, v) := b|A|−1a|B|−1
∑
β′∈B
∏
β′′ 6=β′
∏
α′′∈A
(vβ′′ − uα′′)
(vβ′ − vβ′′)
,
and
J̃(u, v) := a|B|−1b|A|−1
∑
α′∈A
∏
α′′ 6=α′
∏
β′′∈B
(uα′′ − vβ′′)
(uα′ − uα′′)
,
mailto:michel.bauer@ipht.fr
mailto:jean-bernard.zuber@upmc.fr
https://doi.org/10.3842/SIGMA.2020.083
2 M. Bauer and J.-B. Zuber
which is nothing but J with the roles of Px and Qx interchanged. Then:
(i) J and J̃ differ (at most) by a sign, precisely J(u, v) := (−)|A||B|−1J̃(u, v).
(ii) J is a polynomial in (u, v) ∈ RA∪B.
(iii) The δ-identity∫
R
dx δ(Px(u))⊗ δ(Qx(v)) = |J(u, v)|δ(R(u, v)), (1)
(involving J as the multiplier) holds, where for each x ∈ R, δ(Px(u)) and δ(Qx(v)) are
positive σ-finite measures on RA and RB respectively and δ(R(u, v)) is a positive σ-finite
measure on RA∪B = RA × RB.
The main point is the combinatorial identity equation (1) to which we refer as the δ-identity
in the sequel. Before we can prove Proposition 1, we need to make sense of the δ functions that
appear on each side of equation (1). This is the goal of the next section where they are defined
as positive σ-finite measures. First, we make a remark on our motivations.
Remark 1. The δ-identity is grounded in integral geometry but it proves useful in the following
context occurring in probability theory or in statistical mechanics and which we describe infor-
mally using a physicist’s notation. Consider a random vector X = (X1, X2, . . . , Xd), taking its
values in a compact convex set K ⊂ Rd, with a density µ(x2, . . . , xd) on K which is independent
of x1; consider two functions P and Q on K that are polynomial in x1. The joint probability
density function (PDF) of P (X) and Q(X) may be written as
ρ(p, q) = E(δ(P (X)− p)δ(Q(X)− q)). (2)
We rewrite equation (2) as
ρ(p, q) =
∫
dx2 · · · dxd µ(x2, . . . , xd)
∫
dx1 δ(P (x)− p)δ(Q(x)− q)1x∈K . (3)
Suppose that for given (x2, . . . , xd) ∈ Rd−1 either the interval {x1 ∈ R, (x1, x2, . . . , xd) ∈ K} is
empty, or it contains all the roots of the two polynomials P (x)−p and Q(x)−q of the variable x1.
Under this assumption, first one can dispense with the explicit mention of 1x∈K in equation (3)
and then second one can use the identity equation (1) to write the resulting x1 integral over R
in terms of δ(R), where R is the resultant of P (x)− p and Q(x)− q in the variable x1, and of J ,
the expression defined in Proposition 1.
This situation is encountered in the following particular case of Horn’s problem. Consider
two 3 × 3 real traceless symmetric matrices A and B, acted upon by conjugation by matrices
of SO(3). We take A and B as random variables, independently and uniformly distributed on
their orbits characterised by their eigenvalues α = (α1, α2, α3) and β = (β1, β2, β3). Then the
characteristic polynomial of their sum C = A+B is
det(zI3 − C) = det
(
zI3 − diag(α)−R diag(β)RT
)
= z3 + zP (R) +Q(R),
where R ∈ SO(3) is taken randomly and uniformly according to the normalised Haar measure.
Following equation (2), we write the PDF of P and Q in the form
ρ(p, q) =
∫
DRδ(P (R)− p)δ(Q(R)− q).
For example, if R is expressed in terms of Euler angles (θ, φ, ψ), a simple calculation shows
that P (R) and Q(R) are (degree 2) polynomials in c := cos θ, and
ρ(p, q) =
1
2π2
∫ π
0
dφ
∫ π
0
dψ
∫ 1
−1
dc δ(P (c, φ, ψ)− p)δ(Q(c, φ, ψ)− q).
On Products of Delta Distributions and Resultants 3
This is precisely where identity equation (1) simplifies things a great deal, reducing the previous
integral to a (φ, ψ)-integral of δ(R(φ, ψ))J , where R is the resultant of the two polynomials P−p
and Q − q, and J is the Jacobian given above in Proposition 1. One of the two integrations
over φ, ψ localizes on the zeros of R belonging to the appropriate interval, and one is left with
a single integral over the remaining variable. This method has been used in [1] to analyse the
divergences that (somewhat unexpectedly) arise in the PDF ρ(p, q). We refer the reader to
that article for further details. That calculation, carried out in the simple case of two degree 2
polynomials, was the source of inspiration for the more general formula equation (1).
2 Preliminaries
Mainly to fix notation, we recall the definition of the δ measure.
Definition 1. The measure δx, the unit mass concentrated at point x ∈ Rn, is δx(M) := 1x∈M
for M an arbitrary Borel subset of Rn.
Remark 2. An immediate consequence is that if x ∈ Rn and ϕ : Rn → R is any Borel function,
then
∫
Rn δxϕ = ϕ(x). Henceforth, if f : Rn → R is any Borel function, fδx = f(x)δx where the
left-hand side is the product of the measure δx by the function f and the right-hand side the
product of the measure δx by the constant f(x).
The measure δx is a positive σ-finite measure. Moreover, the measure of any bounded set
is finite. Thus δx also defines a distribution: if ϕ : Rn → R is C∞ with compact support, then
〈δx, ϕ〉 := ϕ(x), where the bracket can be interpreted as the integral of ϕ against the measure δx.
This is of course the definition of the δ distribution found in any textbook.
The support of δx is a single point. Our aim is to define positive σ-finite measures whose
support is a finite union of affine hyperplanes in Rn and which are natural extensions of δx. We
do this in three steps. Our definition parallels closely the construction in [2]. We shall briefly
comment on the similarities and differences in Remark 5.
– First step: we define δ(f) when f : R→ R is a regular function. The one dimensional case
is mainly for motivation and also facilitates later comparison with the construction in [2].
Definition 2. A function f : R→ R is called regular if f is C1 and f , f ′ have no common zero.
Definition 3. Let f : R→ R be a regular function. The measure δ(f) is defined by
δ(f) :=
∑
x∈R, f(x)=0
1
|f ′(x)|
δx.
If f : R → R is regular, the number of its zeros in any bounded set is finite (else the set of
zeros would accumulate at a finite point, where necessarily f = f ′ = 0) and the derivative is
nonzero there. Thus δ(f) is a positive σ-finite measure such that the measure of any bounded
Borel set is finite. Hence δ(f) can also be seen as a distribution: if ϕ : R → R is C∞ with
compact support, then 〈δ(f), ϕ〉 :=
∑
x∈R, f(x)=0
1
|f ′(x)|ϕ(x), a finite sum in fact (but the number
of terms may depend on the support of ϕ).
Remark 3. If θk : R → R, k ∈ N is a sequence of Borel functions such that the sequence
of measures θk(t) dt converges weakly to δ0 (that is, if lim
k→+∞
∫
R ϕ(t)θk(t) dt = ϕ(0) for every
continuous function ϕ : R→ R with compact support), and if fa,x : R→ R, t 7→ a(t− x) where
x ∈ R and a ∈ R∗ (the condition under which fa,x is regular), then δ(fa,x) = 1
|a|δx is the weak
limit of the sequence of measures θk(fa,x(t)) dt = θk ◦ fa,x(t) dt. This explains the presence of
the denominators in the definition of δ(f) and motivates that δ ◦f would also be an appropriate
notation.
4 M. Bauer and J.-B. Zuber
The following simple result plays a crucial role:
Lemma 1. If f, g : R→ R are regular functions and the loci f = 0 and g = 0 do not intersect,
then h = fg is regular and
δ(fg) =
1
|f |
δ(g) +
1
|g|
δ(f). (4)
More generally if f1, . . . , fk are regular functions with no pairwise common zeros then the product
f1 · · · fk is regular and
δ(f1 · · · fk) =
k∑
j=1
1∣∣∣ ∏
i 6=j
fi
∣∣∣δ(fj). (5)
Proof. For the first formula, the term 1
|f |δ(g) on the right-hand side for instance is well defined
as the product of a measure δ(g) by a measurable function 1
|f | which is finite in a neighborhood of
the support of δ(g). Then the validity of the first formula is a consequence of (fg)′(x) = f(x)g′(x)
when x is a zero of g, and of Remark 2. The second formula can be justified in a similar way or
from the first by recursion. �
– Second step: we define δ(f) for non-constant affine functions in arbitrary dimensions.
Definition 4. If f : Rn → R is a non-constant affine function then there is a non-constant linear
form l1 6= 0 on Rn and a constant x ∈ R such that f = l1 − x. One can find n− 1 other linear
forms l2, . . . , ln such that (l1, . . . , ln) is a basis of linear forms on Rn. Consequently z : Rn → Rn,
y := (y1, . . . , yn) 7→ z(y) where z(y) := (l1(y), . . . , ln(y)) defines a non-singular linear change of
coordinates. If M is a Borel subset of Rn, we define M̃ ⊂ Rn−1 by z(M)∩{x}×Rn−1 =: {x}×M̃ .
We define the measure δ(f) by
δ(f)(M) :=
∫
M̃
| Jac(y, z)|dz2 · · · dzn,
where Jac(y, z) := det
( ∂yi
∂zj
)
ij
= 1/ Jac(z, y) is the standard Jacobian, a constant that could be
pulled out of the integral.
The fact that the definition of δ(f) is intrinsic, i.e., that δ(f)(M) is independent on how l1
is completed into a basis of linear forms, is guaranteed by the multiplicativity of the Jacobian
determinant. We shall compare this definition of δ(f) with its analog defined by Gelfand and
Shilov in the context of generalized functions in Remark 5. For the moment, let us simply
observe that δ(f) is invariant under translations along the hyperplane {l1 = 0} and its support
is {f = 0}. Moreover, if n = 1 we may write f = fa,x/a as in Remark 3 and a simple computation
shows that the definitions of δ(f) in Definitions 3 and 4 coincide.
An immediate consequence of the definition is that if ϕ : R→ R is a positive Borel function
then ∫
Rn
δ(f)ϕ =
∫
Rn−1
ϕ̃(x, z2, . . . , zn)| Jac(y, z)|dz2 · · · dzn,
where ϕ̃ := ϕ ◦ z−1.
If M is a bounded Borel subset of Rn, then M̃ is also bounded, so the positive measure δ(f)
is such that δ(f)(M) is finite. Hence δ(f) can also be seen as a distribution: if ϕ : R→ R is C∞
with compact support, then
〈δ(f), ϕ〉 :=
∫
Rn−1
ϕ̃(x, z2, . . . , zn)| Jac(y, z)|dz2 · · · dzn, (6)
where ϕ̃ := ϕ ◦ z−1 is again C∞ with compact support.
On Products of Delta Distributions and Resultants 5
If f : Rn → R is a non-constant affine function and a ∈ R∗ then af is a non-constant affine
function and δ(af) = 1
|a|δ(f).
– Third and final step: we define δ(f) for certain products of non-constant affine functions.
Definition 5. Let H(Rn) be the space of functions f : Rn → R that can be written as a product
f = f1 · · · fk of non-constant, pairwise non-proportional, affine functions f1, . . . , fk from Rn
to R.
We say that (f1, . . . , fk) is a factorization of f ∈ H(Rn).
We note that generically fi and fj , i 6= j have common zeros (an affine subspace of dimension
n − 2 unless the 0 loci are parallel hyperplanes) so that {f = 0} is not a smooth submanifold
of Rn.
Definition 6. Suppose that f ∈ H(Rn) and let (f1, . . . , fk) be a factorization of f . We define
δ(f) :=
k∑
j=1
1∣∣∣ ∏
i 6=j
fi
∣∣∣δ(fj)
with δ(fj) as in Definition 4.
It is readily seen that the right-hand side in the definition is still a positive measure on Rn,
and it is σ-finite because
{∏
i 6=j fi = 0
}
is a finite union of affine subspaces of dimension n− 2
in the hyperplane fj = 0 which is the support of δ(fj).
We have to check that δ(f) is defined intrinsically:
Lemma 2. For f ∈ H(Rn), the right-hand side in the definition of δ(f) does not depend on the
factorization of f .
Proof. If (g1, . . . , gl) is another factorization of f ∈ H(Rn) then l = k, there is a unique
permutation σ of J1, kK and unique λ1, . . . , λk ∈ R∗ such that gσ(j) = λjfj for j = 1, . . . , n.
Moreover λ1 · · ·λk = 1. We observe that for each j ∈ J1, kK
1∣∣∣ ∏
i 6=σ(j)
gi
∣∣∣δ(gσ(j)) =
1∣∣∣ ∏
i 6=j
fi
∣∣∣δ(fj)
because δ(λjfj) = 1
|λj |δ(fj) and λ1 · · ·λk = 1, establishing the claim that δ(f) is defined intrin-
sically. �
The space H(R) is simply the space of non-constant real polynomials whose roots are real
and simple. Thus in dimension 1, the condition that the affine functions in the factorization be
non-proportional is equivalent to the condition that f be regular. It is immediate to check that
for f ∈ H(R), the definitions of δ(f) in Definitions 3 and 6 coincide.
If f , g and fg belong to H(Rn), it is immediate to check that
δ(fg) =
1
|f |
δ(g) +
1
|g|
δ(f).
which is formally identical to equation (4). The extension to a product of a finite number of
factors leads to a counterpart of equation (5).
In fact, once δ(f) is defined when f is a non-constant affine function, the definition we have
given of δ(f) for f ∈ H(Rn) could be seen as a formal extension of the identity equation (5) in
this more general context. However, there is one important difference between the two situations:
6 M. Bauer and J.-B. Zuber
If f ∈ H(Rn), the δ(f)-measure of a compact set is clearly finite if it does not contain common
zeros of fi and fj for any i 6= j, but may be infinite otherwise. Similarly, positive C∞ functions
with compact support may have an infinite integral against δ(f). As an illustration, take n = 2,
f(y) = y1y2, and ϕ : R2 → R a C∞ function, positive with compact support and such that
ϕ(0, 0) > 0 (i.e., ϕ does not vanish in a neighborhood of the intersection y1 = y2 = 0). Then∫
R2
δ(f)ϕ =
∫
R
ϕ(0, y2)
dy2
|y2|
+
∫
R
ϕ(y1, 0)
dy1
|y1|
= +∞.
This is the reason why for a generic f ∈ H(Rn), n > 1, the measure δ(f) cannot be interpreted
as a distribution and distribution theory on Rn cannot be the natural setting.
It will be clear from the discussion in Remark 5 below that a positive σ-finite measure δ(f)
can be defined for more general f ’s, but each of the functions Px, Qx, R involved in equation (1)
is the product of pairwise non-proportional affine functions on some Rn, so our construction
is enough to make sense of δ(Px), δ(Qx), δ(R), and we have achieved the preliminary goal of
making sense of both sides of the δ-identity equation (1).
We apply the definitions to an explicit computation illustrating their naturalness. In what
follows, a is always a fixed parameter. Recall that Px(u) := a
∏
α∈A(x − uα), a notation sug-
gesting that x is a (real) parameter and u ∈ RA is the variable. But we could also denote the
same expression by Pu(x), a notation suggesting that u is a (vector) parameter and x ∈ R is the
variable. Finally, we could denote the same expression by P (x, u), a function on R × RA. No
matter how we split between parameters and variables,
∏
α∈A(x− uα) is a product of |A| affine
functions. We compare δ(Px), δ(Pu) and δ(P ).
– The function Px is a product of |A| pairwise non-proportional affine functions of u ∈ RA.
We may use (uα′ − xδα,α′)α′∈A as coordinates near the zero locus of x− uα. We infer from the
definitions that, if ϕ is a positive Borel function on RA,∫
δ(Px)ϕ =
1
|a|
∑
α∈A
∫
RA\{α}
1∏
α′ 6=α
|x− uα′ |
ϕ(u)|uα=x
∏
α′ 6=α
duα′ . (7)
This equation (and its counterpart for Qx) will be the starting point of our proof of the δ-identity
equation (1).
– The polynomial (in x) Pu(x) is regular if and only if the uαs are pairwise distinct, which
is also the condition under which the affine functions fα(x) := x − uα of x are pairwise non-
proportional and all definitions of δ(Pu) lead to∫
δ(Pu)ϕ =
1
|a|
∑
α∈A
1∏
α′ 6=α
|uα − uα′ |
ϕ(uα) =
∑
α∈A
1
|P ′u(uα)|
ϕ(uα)
for any positive Borel function ϕ on R, where P ′u denotes the derivative of Pu with respect to x.
– The function P is a product of |A| pairwise non-proportional affine functions on R × RA.
We infer from the definitions that if ϕ is a positive Borel function on that product space∫
δ(P )ϕ =
1
|a|
∑
α∈A
∫
RA
1∏
α′ 6=α
|uα − uα′ |
ϕ(uα; (uα′)α′∈A)
∏
α′∈A
duα′ ,
or equivalently∫
δ(P )ϕ =
1
|a|
∑
α∈A
∫
R×RA\{α}
1∏
α′ 6=α
|x− uα′ |
ϕ(x; (uα′)α′∈A)|uα=xdx
∏
α′ 6=α
duα′ .
The first formula emerges if one uses (x−uα; (u′α)α′∈A) while the second formula emerges if one
uses (x; (u′α − xδα,α′)α′∈A) as coordinates on R× RA near the zero locus of x− uα.
On Products of Delta Distributions and Resultants 7
Remark 4. It is clear that all these formulæ are pretty much one and the same, though the one
for Pu involves the restriction that the components of the vector u be pairwise distinct (unless
we are prepared to deal with non σ-finite measures).
We finally turn to a comparison of the construction given above with the one in [2]. The
following cursory discussion is used nowhere in the sequel and the uninterested reader can jump
directly to Section 3. The discussion involves the generalization of the notion of regular function
for functions from Rn to R.
Definition 7. A function f : Rn → R is called regular if f is C1 and its gradient Grad f vanishes
nowhere on f = 0.
Remark 5. In [2], Gelfand and Shilov define δ(f) (which we denote by δGS to make later
comparison with our definitions easier and avoid confusion) as a generalized function when
f : Rn → R is an arbitrary regular function. We have already stressed that this is not an auto-
matic consequence of the definition of δ0 (either as a measure or as a distribution). However
(maybe modulo the inclusion of absolute values of Jacobians, see the first point below) the defi-
nition chosen by Gelfand and Shilov is the “only” natural one, the one that can be manipulated
most intuitively. For instance, it behaves nicely if some variables in f are treated as parameters.
We have seen an instance of this in our context when we compared δ(Px), δ(Pu) and δ(P ).
Another view of the naturalness properties of the definition of δGS(f) is that if θk : R → R,
k ∈ N is a sequence of smooth functions converging towards δ0 in the distribution topology (on R)
then the compositions θk(f) = θk ◦f converge towards δ(f) in the distribution topology (on Rn).
We have seen a trivial counterpart of this in Remark 3. The “approximation of distribution
by smooth maps” procedure is in fact the path followed by Hörmander in [3, Chapter 6 and
Section 8.2] to define the composition of a general distribution on Rm with functions from Rn
to Rm satisfying appropriate conditions (surjectivity of the differential everywhere in Chapter 6,
less stringent conditions allowing certain singularities in Section 8.2).
The reader is invited to consult [2] for the detailed definition of δGS and other distributions
localized of f = 0. Coming back to the comparison, one can check the following:
– First, in one dimension, δGS(f) is the distribution associated to the measure δ(f) from
Definition 3, with one little proviso: Gelfand and Shilov do not include an absolute value for
the derivatives in the denominators. More generally, they do not include absolute values for
Jacobians because they use the framework of differential forms, whereas densities better fit our
needs. When we talk of δGS in the sequel, we always have in mind that absolute values are
included.
– Second, an affine function f : Rn → R is regular if and only if it is non-constant, and
then δGS(f) is the distribution associated to the measure δ(f) from Definition 4. In fact, the
definition of δGS(f) for a general regular function follows the same pattern, using localization
and replacing the linear change of coordinates by the implicit function theorem. Borrowing
the notations from Definition 4, z1 = f(y), z2, . . . , zn become arbitrary local coordinates, the
Jacobian is not a constant anymore and the formula for 〈δGS(f), ϕ〉 is similar to equation (6),
but with Jac(y, z) replaced by Jac(y, z)z1=0. Using a partition of unity, one may assume that the
local coordinates are well-defined in an open set containing the support of ϕ so that ϕ̃ = ϕ◦ z−1
extends as a C∞ function with compact support. The consistency of the procedure is guaranteed
by the general change of variable formula. We refer again to [2] for all details.
– Third, the identity equation (4) (with δGS substituted for δ, and interpreted as an identity
between distributions) remains true whenever g, f, fg : Rn → R are regular. Then the analog
of equation (5) also holds. Thus, if f ∈ H(Rn) is regular, δGS(f) is the distribution associated
to the measure δ(f) from Definition 6. It is immediate that an element of H(Rn) is regular if
and only if all factors in a factorization have the same linear part but pairwise distinct constant
8 M. Bauer and J.-B. Zuber
terms. As already noticed, all members of H(R) are regular, but generic members of H(Rn),
n > 1, are not. For such members f , the measure δ(f) assigns an infinite measure to certain
bounded Borel subsets of Rn, and δ(f) cannot be interpreted as a distribution.
– Fourth, however, every f ∈ H(Rn) with factorization (f1, . . . , fk) is regular on a dense
connected open subset of Rn, namely the complement Of of ∪i 6=j{fi = fj = 0}. The Gelfand–
Shilov construction defines a distribution (which we still denote by δGS(f)) on Of , which is
positive on positive test functions (with support in Of by definition), hence defines a positive
σ-finite measure on Of , and on Of equation (5) again holds. Hence this measure coincides
with δ(f) (as defined in Definition 6) on Borel subsets of Of . Consequently δ(f) can also be
interpreted as an extension of (the positive σ-finite measure on Of associated to) δGS(f) to the
whole of Rn. The ambiguity is a positive σ-finite measure with support in ∪i 6=j{fi = fj = 0},
and Definition 6 consists in extending by the zero measure. We shall not try to explore whether
some other extensions would preserve the nice properties of the definition of δ(f), in particular
whether some generalized version of equation (1) could be obtained in that way. Finally we note
that extension by the zero measure could be used much more generally to extend a (positive
σ-finite) measure defined on some Borel subset of a measurable space to the whole space, as is
customary done for the similar construction of extensions for distributions.
3 Proof of Proposition 1
Proof. To avoid clumsy notation, we write simply P (x) (resp. Q(x)) for Pu(x) (resp. Qv(x))
in this proof.
We start with the proof of (i). Recall that
J(u, v) = a|B|−1b|A|−1
∑
β′∈B
∏
β′′ 6=β′
∏
α′′∈A
(vβ′′ − uα′′)
(vβ′ − vβ′′)
.
Rewrite
a|B|−1b|A|−1
∏
β′′ 6=β′
∏
α′′∈A
(vβ′′ − uα′′)
(vβ′ − vβ′′)
= b|A|−1
∏
β′′ 6=β′
P (vβ′′)
(vβ′ − vβ′′)
=
b|A|
P (vβ′)Q′(vβ′)
∏
β′′∈B
P (vβ′′)
and notice that if S(x) := P (x)Q(x), then P (vβ′)Q
′(vβ′) = S′(vβ′) to conclude that
J(u, v) = b|A|
∑
β′∈B
1
S′(vβ′)
∏
β′′∈B
P (vβ′′) = b|A|a|B|
∑
β′∈B
1
S′(vβ′)
∏
α′′∈A
β′′∈B
(vβ′′ − uα′′). (8)
The same manipulation on
J̃(u, v) = a|B|−1b|A|−1
∑
α′∈A
∏
α′′ 6=α′
∏
β′′∈B
(uα′′ − vβ′′)
(uα′ − uα′′)
leads to
J̃(u, v) = a|B|
(∑
α′′∈A
1
S′(uα′)
) ∏
α′′∈A
Q(vα′′) = a|B|b|A|
(∑
α′′∈A
1
S′(uα′)
) ∏
β′′∈B
α′′∈A
(uα′′ − vβ′′). (9)
On Products of Delta Distributions and Resultants 9
But if S = c
∏
γ∈C(x−wγ) is an arbitrary polynomial of degree at least 2 with simple zeros, one
has the “well known” identity1
∑
γ∈C
1
S′(wγ)
= 0. For the polynomial S at hand this entails
∑
α′∈A
1
S′(uα′)
= −
∑
β′∈B
1
S′(vβ′)
.
Comparison of equations (8) and (9) leads immediately to J = (−)|A||B|−1J̃ concluding the proof
of (i).
To prove (ii) we simply note that the original formula for J is a polynomial in u with rational
coefficients in v while the original formula for J̃ is a polynomial in v with rational coefficients
in u. As J and J̃ differ (at most) by a sign, altogether J is a polynomial in (u, v).
We turn finally to the proof of (iii). We rewrite equation (7) in a number of more compact
forms, each of which we shall use freely, namely
δ(Px) =
1
|a|
∑
α∈A
1∏
α′ 6=α
|x− uα′ |
δx(uα)
∏
α′ 6=α
duα′ =
∑
α∈A
1
|P ′(uα)|
δx(uα)
∏
α′ 6=α
duα′
=
1
|a|
∑
α∈A
1∏
α′ 6=α
|x− uα′ |
δ(uα − x) =
∑
α∈A
1
|P ′(uα)|
δ(uα − x).
In the first line, δx is the one-dimensional δ function for a unit mass at point x along the
coordinate axis α, while in the second line δ(uα − x) is a distribution on RA, namely δ(f) for
the affine function f : RA → R, u 7→ f(u) := uα − x. Of course, the definition of δ(f) yields
δ(uα − x) = δx(uα)
∏
α′ 6=α duα′ . To get the second equality in each line, we have used that
a
∏
α′ 6=α(x − uα′) and a
∏
α′ 6=α(uα − uα′) = P ′u(uα) are equal on the support of δx(uα) or of
δ(uα − x).
Now δ(Px)δ(Qx), which could be rewritten more carefully as δ(Px) ⊗ δ(Qx), is well defined
as the product of two measures, δ(Px) on RA and δ(Qx) on RB. When P and Q are monic of
degree 1, P = (x− u), Q = (x− v),∫
dx
∫
ϕ(u, v)δx(u)δx(v) =
∫
dxϕ(x, x)
holds for every positive Borel function ϕ on R2, and the very definition of δ(u− v) as a positive
measure on R2 ensures that
∫
dx δx(u)δx(v) = δ(u − v). Thus (iii) holds in this special case,
and we shall use it to prove the general case. Namely we translate the special case∫
x
dx δx(uα)δx(vβ) = δ(uα − vβ),
where δ(uα − vβ) is to be interpreted as the measure in the plane (uα, vβ) ∈ R2, into∫
dx δ(uα − x)δ(vβ − x) = δ(uα − vβ),
where now δ(uα − x) is interpreted as the measure δ(f) on RA for f(u) = uα − x, δ(vβ − x) is
interpreted as the measure δ(g) on RB for g(v) = vβ − x, and δ(uα − vβ) as the measure δ(h)
on RA∪B for h(u, v) = uα − vβ. Thus∫
dx], δ(Px)δ(Qx) =
∑
α∈A
β∈B
∫
dx
1
|P ′(uα)|
δ(uα − x)
1
|Q′(vβ)|
δ(vβ − x)
1For instance, note that 1
S(x)
=
∑
γ∈C
1
S′(wγ)
1
x−wγ
on the one hand, and 1
S(x)
= 0
(
1/x2
)
at infinity.
10 M. Bauer and J.-B. Zuber
=
∑
α∈A
β∈B
1
|P ′(uα)||Q′(vβ)|
δ(uα − vβ) (10)
=
∑
α∈A
β∈B
1
|P ′(vβ)||Q′(uα)|
δ(uα − vβ), (11)
where in the last line we have used that 1
P ′(uα)Q′(vβ)
and 1
P ′(vβ)Q′(uα)
coincide on the support Σα,β
of the measure δ(uα − vβ). On the other hand, using Definition 6 for R(u, v) we get
δ(R) =
∑
α∈A
β∈B
1
|Jα,β(u, v)|
δ(uα − vβ), (12)
where
Jα,β(u, v) := a|B|b|A|
∏
(α′,β′)6=(α,β)
(uα′ − vβ′).
Observe that, on Σα,β,
a
∏
α′ 6=α
(uα′ − vβ) = (−)|A|−1P ′(vβ), b
∏
β′ 6=β
(uα − vβ′) = Q′(uα),
so that, again on Σα,β,
Jα,β(u, v) = a|B|−1b|A|−1
∏
α′′ 6=α
β′′ 6=β
(uα′′ − vβ′′)(−)|A|−1P ′(vβ)Q′(uα). (13)
Fix α ∈ A and β ∈ B. If β′ 6= β then
∏
β′′ 6=β′
∏
α′′∈A(vβ′′−uα′′) contains a factor vβ−uα. Thus,
if vβ = uα, all terms but the one corresponding to β′ = β in the sum defining J(u, v) vanish.
Hence, on Σα,β, we have
J(u, v) = a|B|−1b|A|−1
∏
β′′ 6=β
∏
α′′∈A
(vβ′′ − uα′′)
(vβ − vβ′′)
= a|B|−1b|A|−1
∏
β′′ 6=β
∏
α′′∈A
(vβ′′ − uα′′)
(uα − vβ′′)
= a|B|−1b|A|−1(−)|B|−1
∏
α′′ 6=α
β′′ 6=β
(vβ′′ − uα′′).
Using equation (13), we obtain that, on Σα,β,
|J(u, v)|
|Jα,β(u, v)|
=
1
|P ′(vβ)||Q′(uα)|
.
Comparison of equations (11) and (12) establishes the validity of (iii). �
Remark 6. The explicit expression of the polynomial J(u, v) is quite complicated in general.
It is rather simple in the case when Q is of degree 2. Then, writing v = (v′, v′′) i.e., Qv(x) =
b(x− v′)(x− v′′) one obtains
J(u, v) = −b|A|−1P (v′)− P (v′′)
v′ − v′′
,
which is a simple divided difference, hence is obviously polynomial in (u, v). Even the case
when P and Q are of degree 3 leads to a prohibitively complicated explicit expression for J(u, v)
as a polynomial.
Remark 7. It is a simple consequence of equation (10) that integration over x does not introduce
unexpected singularities: the measure of a compact set K in RA∪B under
∫
dx δ(Px)δ(Qx) is
finite if K does not meet any hyperplane uα = uα′ , α 6= α′ or vβ = vβ′ , β 6= β′.
On Products of Delta Distributions and Resultants 11
Acknowledgements
We thank Michel Talagrand for his comments and encouragement.
References
[1] Coquereaux R., Zuber J.-B., The Horn problem for real symmetric and quaternionic self-dual matrices,
SIGMA 15 (2019), 029, 34 pages, arXiv:1809.03394.
[2] Gelfand I.M., Shilov G.E., Generalized functions, Vol. I, Properties and operations, Academic Press, New
York – London, 1964.
[3] Hörmander L., The analysis of linear partial differential operators, Vol. I, Distribution theory and Fourier
analysis, Grundlehren der Mathematischen Wissenschaften, Vol. 256, Springer-Verlag, Berlin, 1983.
https://doi.org/10.3842/SIGMA.2019.029
https://arxiv.org/abs/1809.03394
https://doi.org/10.1007/978-3-642-96750-4
1 Introduction
2 Preliminaries
3 Proof of Proposition 1
References
|
| id | nasplib_isofts_kiev_ua-123456789-210765 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-15T14:40:57Z |
| publishDate | 2020 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Bauer, Michel Zuber, Jean-Bernard 2025-12-17T14:30:59Z 2020 On Products of Delta Distributions and Resultants. Michel Bauer and Jean-Bernard Zuber. SIGMA 16 (2020), 083, 11 pages 1815-0659 2020 Mathematics Subject Classification: 46F10; 49Q15 arXiv:2006.08301 https://nasplib.isofts.kiev.ua/handle/123456789/210765 https://doi.org/10.3842/SIGMA.2020.083 We prove an identity in integral geometry, showing that if ₓ and ₓ are two polynomials, ∫dxδ( ₓ) ⊗ δ( ₓ) is proportional to δ( ) where is the resultant of ₓ and ₓ. We thank Michel Talagrand for his comments and encouragement. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications On Products of Delta Distributions and Resultants Article published earlier |
| spellingShingle | On Products of Delta Distributions and Resultants Bauer, Michel Zuber, Jean-Bernard |
| title | On Products of Delta Distributions and Resultants |
| title_full | On Products of Delta Distributions and Resultants |
| title_fullStr | On Products of Delta Distributions and Resultants |
| title_full_unstemmed | On Products of Delta Distributions and Resultants |
| title_short | On Products of Delta Distributions and Resultants |
| title_sort | on products of delta distributions and resultants |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/210765 |
| work_keys_str_mv | AT bauermichel onproductsofdeltadistributionsandresultants AT zuberjeanbernard onproductsofdeltadistributionsandresultants |