Compact factors and homogeneous equations

Original edition by GPT-6.1 Sol (OpenAI), Ultra reasoning effort. Repaired and self-checked by GPT-6 Astra (OpenAI), Ultra reasoning effort, October 2026. Original exposition: public domain (CC0); supplied prerequisites retain their stated licences.

Complex Fourier zeros determine which compact distributions on the line admit nontrivial convolution factors. We classify every distribution whose compact factorizations are trivial and construct an actual compact factor from any complex zero. In several dimensions, homogeneity makes polynomial null solutions dense in every smooth global null solution. The same polynomial moment condition characterizes compactly supported solutions of the forced equation. We prove the entire quotient convergence, its growth and its compact inverse transform, including complex coefficients, repeated factors and zero or constant symbols. Finite complex zero sets characterize point jets, compact solutions have the exact forcing hull, and polynomial approximations can preserve any finite compatible observations.

Our convention is , , and bilinear complex-linear distributional pairings. Convolution as addition of supports, B0–B3 and Theorem1.1, supplies compact actions, locality, parameter pairing, convolution and differentiation. Compact forcing, moments and positive error kernels, Lemma1.2 and Theorem1.3, proves polynomial-moment detection and compact primitives. Fundamental solutions, continuation and approximation, B1–B2 and Lemma3.1, supplies smooth completeness, compact duality and separation, using the supplied functional foundation, §5, for complex Hahn–Banach.

Cauchy kernels and distributional boundary limits, Theorem1.1 and Corollary2.2, supplies Cauchy integration, Taylor series and coefficient estimates. Boundary flux and weak identities, Corollaries2.3–2.4, proves complex Green and its finite-corner form. The finite-order construction in the proof of Sharp bounds for compact spectra, Theorem3.1, applies to any compact distribution; its Lemma0.1 supplies the identity principle. The supplied Fourier foundation, F1–F5, proves inversion and coordinate rules. The supplied finite algebra, §§10.1–10.5, proves basis extension, elimination and orthogonal coordinates; scalar, §§12–13, and integration, §§15–16, foundations supply the calculus, convergence and substitution used below. We prove the needed polynomial factorization next.

Polynomial roots from the Cauchy estimate

Lemma 0.1 (complete one-variable factorization). A nonzero complex polynomial of degree is its leading coefficient times a product of linear factors, with every root repeated according to its multiplicity. For the product is empty.

Proof. First, a bounded entire function , with , is constant. At any centre , the Cauchy coefficient estimate from U013, Corollary2.2, on every circle of radius gives . Let ; all first derivatives vanish. Integrating along each line segment by the scalar fundamental theorem shows that is constant on the plane.

Suppose , , has no zero. Its reciprocal is entire by the quotient derivative rule and U013, Corollary2.2. Since as , choose such that outside that disk. On the disk, continuity and nonvanishing give a positive minimum of . Thus is bounded on the whole plane and tends to zero at infinity. The preceding argument makes it identically zero, contradicting . Hence a root exists.

The finite telescoping identity divides out that root and leaves a degree- polynomial with leading coefficient . Induction completes the factorization. At any root, the number of its occurrences in the product is exactly the first nonzero Taylor order, because all other factors are nonzero there. This proves the multiplicity statement and also the bound of on the number of distinct roots.

Every compact distribution with only trivial convolution factors

Call a factor trivial when it is a nonzero scalar multiple of a shifted point mass. A factorization is trivial when at least one of its factors is trivial. These are the units of the compact-distribution convolution algebra: has inverse . Treating only the mass-one as trivial would make scalar refactorizations of every nonzero input fail the intended condition; the algebraic unit convention makes that distinction explicit.

Theorem 1.1. A nonzero has only trivial convolution factorizations , , if and only if

Every zero-free compact transform is a point mass. For , its entire Fourier transform is justified by the same finite-order exponential-series construction as in Sharp bounds for compact spectra, formula (3.2), with the exponent sign changed and the inverse-transform normalization omitted. That argument uses only compact finite order for its entire construction and test interchange; the later norm estimates of that theorem are not required here. A compact cutoff supported in some and the finite-order estimate give The real restriction is the whole tempered Fourier transform, by the absolute finite-order test interchange already proved in Sharp bounds for compact spectra, Theorem 3.1. If , Fourier injectivity gives .

Suppose has no complex zeros. The entire function has, by Cauchy kernels and distributional boundary limits's Cauchy coefficients, a Taylor series about zero that converges on every disk. Integrating that series termwise gives an entire primitive with . Choose a complex number with , using its nonzero polar form, and put . Differentiation shows that is constant and its value at zero is one, so on the whole plane.

Write . Formula (1.2) implies For each , denote the right side at radius by . On the circle , the real function is nonnegative. Uniform convergence of the Taylor series on this circle, obtained from any larger Cauchy circle, gives its mean . For , the -th positive Fourier coefficient of is . Consequently For every , let ; the right side is , so . Therefore , with . On the real axis (1.2) is merely a polynomial bound. Applying it in both directions and forces . Write , . Fourier injectivity now gives . This proves the entire zero-free assertion, including the reality of the point location, without an unsupported entire-factorization theorem.

A complex zero gives an actual compact factor. Suppose has a zero at . Set , a compact distribution. Its total mass is . Compact forcing, moments and positive error kernels's complete compact-primitive theorem, with and order one, produces a compact distribution with , supported in any closed interval containing the support of . Put , still compact. The full distributional product rule gives Thus, with , Compact-factor convolution and differentiation are the complete Convolution as addition of supports interfaces. The entire transform of is , so this factor is not a unit. The factor is nonzero because , and its entire transform is precisely the holomorphic quotient , by (1.4). Compactness was proved directly by the moment theorem, rather than asserted from a growth theorem.

If every factorization of is trivial, then either is zero-free, in which case the preceding result gives , or it has a complex zero. In the latter factorization must be a unit, since is not. Write and use (1.4): This is (1.1). It proves the complete necessity.

Sufficiency, including every complex multiplicity. If has the form (1.1), its entire transform is For any compact factorization , neither factor is zero. Their entire transforms multiply to : the compact convolution pairing applied to separates into the two finite-order pairings. If , (1.5) has no zeros, so both factor transforms are zero-free. If , it has exactly one zero, of multiplicity one. Both factor transforms cannot have a zero: different zero locations would give two distinct zeros of the product, while the same location would give multiplicity at least two. The multiplicities add because each local Taylor series factors into its first nonzero power times a nonvanishing analytic function. Hence at least one factor is zero-free. The proved zero-free result makes that compact factor a nonzero scalar shifted point mass. This proves sufficiency in every case. Finally, any unit has a zero-free entire transform because its product with its inverse transform is one; the proved zero-free result and the explicit inverses above show that the scalar shifted point masses are all the units.

When the linear factors give a finite factorization

Theorem 1.1 supplies a factor at each complex zero. A finite list of zeros has a particularly concrete meaning in physical space.

Theorem 1.2 (finite zeros and point jets). For nonzero , the following conditions are equivalent: has finitely many zeros counted with multiplicity; is a finite point jet at one real point; and is a finite convolution of units and the linear nonunit factors in Theorem 1.1. If its zeros are , repeated according to multiplicity, then The zero-free case has and is a unit.

Proof. At an isolated zero, the convergent Taylor series of a nonzero entire function has a first nonzero term of finite degree. The compact factor construction in the preceding proof removes one occurrence of that zero: it gives an actual compact with . The Taylor series shows that the multiplicity decreases by one there, and the quotient creates no other zeros. It is nonzero by this identity. Repeating through the finite list leaves a compact factor whose transform is zero-free. Theorem 1.1's proved zero-free classification makes it , giving (K1) with the original Fourier signs.

If the product polynomial is , the ordinary point derivatives in (K1) have coefficients , because . Conversely a nonzero finite jet has transform . At least one coefficient is nonzero because the given distribution is nonzero; thus that polynomial is nonzero. Lemma0.1 supplies its leading coefficient and every root with multiplicity, so it gives a finite convolution as in (K1). Finally each classified nonunit is a first-order point jet, and finite convolutions of point jets and units remain point jets at the sum of their real support points. This proves all implications.

The forcing , , has zeros for every integer . Each zero supplies a compact linear factor, but Theorem 1.2 excludes a finite factorization solely into the classified linear factors and units. The distinction concerns the complete zero set, including every multiplicity.

Homogeneous equations: compact solvability and smooth density

Theorem 2.1. Let be a homogeneous complex polynomial in variables, and let Then its polynomial members are dense in in the topology of uniform convergence of every derivative on every compact set. For , there exists with if and only if for every polynomial satisfying . Both assertions include the zero polynomial and the nonzero constant case.

We first prove the analytic mechanism and then turn the moment condition into exactly the required quotient.

A uniform division estimate on short complex lines

Let be homogeneous of degree . Choose a real vector with . Such a vector exists: a polynomial zero at every real point is zero, by induction on the number of variables, applying the one-variable finite root bound to each coefficient in the remaining variables. For every , the one-variable polynomial has degree and the fixed leading coefficient . Lemma0.1 writes it as retaining roots with all multiplicities. Set . Remove from the intervals of distance less than from the real numbers . Their total length is at most , so there is outside them. On the entire circle , The radius may depend on ; the lower bound does not.

If with a polynomial or an entire function, the one-variable Cauchy mean formula for gives This is also valid at zeros of , because it applies to the already analytic , without evaluating a singular pointwise quotient there.

In particular, suppose satisfies the whole growth estimate If its quotient is entire, (2.3), and give Only the constant changes: and . Thus division keeps polynomial growth on the real subspace and finite exponential growth in the imaginary directions.

A compact inverse transform from that growth

We prove the exact range statement needed here. Let be any entire function on satisfying (2.5). Its real restriction has polynomial growth, so it is tempered, and is a well-defined tempered distribution. In fact This coordinate bound establishes compactness. The next lemma gives the sharper Euclidean radius with the same exponent.

For , put The integral is absolute, and identifies its function distribution with the inverse transform of , by absolute Fubini against a Schwartz test. As , the latter functions converge to in : the polynomial bound times any Schwartz test is integrable and dominates their differences. Fourier continuity therefore gives distributionally.

Fix a coordinate , a real vector , and a real height . Shift the -th integration line from to . The full identity is Here is the contour justification, including the unbounded integration. Hold the other real coordinates fixed and use a rectangle with vertical sides at and the two horizontal heights . Its integrand is entire in that coordinate. Its closed integral is zero by the complex Green identity from Boundary flux and weak identities, Corollaries2.3–2.4: first round the four corners to apply the smooth-boundary identity, then let their radii tend to zero; continuity makes the four removed arc contributions tend to zero, and the area Cauchy–Riemann term is identically zero. On a vertical side, with height between , the modulus is bounded by a polynomial in times For fixed , its integral over and over the other coordinates tends to zero as ; the Gaussian dominates that polynomial. The horizontal tails also tend to zero by the same majorant. Fubini is absolute in all these bounded-height integrals. This proves (2.7) for each fixed finite , of either sign.

Formula (2.5) in (2.7) gives, with constants independent of , For the integral bound use , then the substitution ; the integral of is finite.

If , choose . If , choose . Both choices make the last exponential in (2.8) equal On every compact subset of , its distance from the box boundary is positive and is bounded. Thus the polynomial factor in (2.8) is bounded by a fixed power of , while this exponential tends to zero faster than any such power. Consequently uniformly there. Distributional convergence now gives on that open coordinate region. The complement of is the finite union of these coordinate regions; local smooth partitions of unity, or the locality of a zero distribution, give (2.6). In particular is a compact distribution. This proves the full inverse-transform mechanism used below.

The unchanged radial exponent gives a Euclidean ball

The coordinate argument also gives the sharper geometric interpretation of the same constant .

Lemma 2.2 (radial support radius). Under (2.5), the inverse transform satisfies

Proof. Fix a real unit vector , and choose a real orthogonal coordinate matrix whose first coordinate direction is . To construct this matrix, extend the nonzero vector to a real basis and apply the proved Gram process in the supplied finite algebra, §§10.1 and10.5, beginning with ; its first normalized vector stays . The resulting matrix satisfies , so determinant multiplicativity gives . Its real Jacobian therefore has absolute value one; it preserves the Euclidean norm and, after complex linear extension, the bilinear quadratic form . In these coordinates the complete rectangle proof of (2.7) shifts the first line by , or equivalently shifts the original frequency vector to , for each fixed .

The vertical sides still have Gaussian decay in their real coordinate and in all remaining real coordinates, multiplied by a polynomial at fixed finite height. Thus their integrals vanish, the horizontal tails converge, and all integrations are absolute exactly as in that proof. Applying the radial bound gives No uniformity of the constant in is needed. On a compact subset of , take . The last exponential is , with a uniform positive gap. It dominates every remaining inverse power of , so uniformly there. The already proved distributional convergence gives on this open half-space.

For any with , take . The strict half-space contains a neighborhood of . Thus that point is outside the support of , proving (K2), including . The exponent was never enlarged.

The moment condition produces an entire quotient

Let annihilate every polynomial in . Its entire Fourier transform has the homogeneous Taylor pieces The compact finite-order estimate proves entire convergence and (2.4), by the entire compact-Fourier construction in Sharp bounds for compact spectra, Theorem 3.1, and the preceding proof: choose a compact cutoff supported in a ball of radius ; derivatives of the exponential give the polynomial factor, and its modulus is at most . Choose a larger if necessary. The real restriction is exactly .

Let be the finite-dimensional space of homogeneous polynomials of degree . The bilinear coefficient pairing is nondegenerate. For , (2.9) says . For , kills all of , so . For , put The finite derivative identity, with all factors retained, is It follows by expanding , and into their monomials: the factorial remaining after differentiation is exactly the coefficient pairing on the right.

The functional vanishes on by the moment hypothesis. It therefore factors through . Extend that finite-dimensional linear functional to by basis extension, and represent it as , using nondegeneracy. Formula (2.10) then gives The quotient polynomial is unique because the complex polynomial ring has no zero divisors: the product of the leading monomials in any fixed lexicographic order has a nonzero coefficient. The scalar factors in (2.9)–(2.11) are never suppressed.

It remains to prove that the formal quotient actually converges everywhere. Apply (2.3) to the polynomial identity (2.11). For , put . On the selected circle, . For every , the one-variable Cauchy formula applied to , , gives Indeed the coefficient of is , its Cauchy bound is , and homogeneity supplies the factor . Thus The right side is a summable geometric sequence. Therefore converges uniformly on every complex compact ball. It is entire: use the same uniform bounds on a slightly larger ball and apply the one-variable Cauchy derivative estimate successively in the coordinates to each polynomial term; every differentiated series is then locally uniformly convergent. Their sums satisfy the coordinate Cauchy–Riemann equations and the corresponding local multivariable power-series expansion. Multiplication by the fixed polynomial passes through the locally uniform series, giving everywhere.

Now (2.3) applies to that entire quotient and (2.4) gives (2.5). The proved inverse-transform statement supplies a compact . On real frequencies, ; the full distributional Fourier differentiation rule and injectivity give . This proves sufficiency in (2.1) for every nonzero homogeneous polynomial of positive degree.

Necessity retains the bilinear transpose sign. If with compact, then for every polynomial , Compact support permits pairing with that global polynomial by a cutoff, and the distributional product and transpose rules give the displayed identity. The homogeneity is what makes . This proves both directions of the exact compact-solvability criterion.

Density in the full smooth topology

Let be the closure in of the vector space of polynomial null solutions. Continuity of makes closed, so . Suppose . The complete compact-distribution separation lemma Fundamental solutions, continuation and approximation, Lemma 3.1 produces with and . In particular it annihilates all null polynomials. The just-proved criterion gives a compact with . But (2.13), now applied to the smooth global null solution , gives a contradiction. Therefore .

This is sequential approximation in exactly the stated topology. For each integer , choose a null polynomial so that Closure in the smooth seminorm topology permits this finite-seminorm choice. Every fixed compact set and derivative order is contained in these controls for all sufficiently large , so with every derivative on every compact set. No bound on the growth of , ellipticity, real coefficients or simplicity of the factors of was assumed.

If is a nonzero constant, its smooth null space and polynomial null space are both zero, the moment condition is empty, and is compact for every . If , all polynomials and all smooth functions are null. A compact annihilating every polynomial is zero by Compact forcing, moments and positive error kernels, Lemma 1.2, so the compact equation has a solution exactly under the stated condition. Polynomials are dense in the full smooth space: otherwise Fundamental solutions, continuation and approximation separation would produce a nonzero compact distribution annihilating them, contradicting that same complete moment-detection lemma. The identical seminorm selection gives sequential approximation in this case too. All degenerate cases are thus included.

Compact uniqueness and the actual support hull

Existence in Theorem 2.1 uses homogeneity. Once a compact solution exists, its uniqueness and support geometry hold for every nonzero polynomial.

Corollary 2.3. Let be any nonzero complex polynomial on . A compact solution of is unique. For nonzero forcing, where denotes the closed convex hull. For zero forcing the unique compact solution is zero.

Proof. Set . Its support is contained in , and its entire transform is the nonzero polynomial , so and its support is exactly . The differentiation and compact convolution rules give . The full compact convolution support theorem in Convex supports and convolution cancellation, Theorem 3.1, applies to arbitrary complex distributions and gives If , its hull is nonempty and the right side is nonempty, so . Apply this to the difference of two compact solutions to get uniqueness, and to a solution of nonzero forcing to get (K4). Nonzero constants are included.

Equality of hulls allows interior support to change. The compact primitive of has interval support, while the forcing has only two support points. This corollary does not extend the polynomial-moment existence criterion to nonhomogeneous symbols.

Approximation that preserves finitely many observations

Smooth convergence controls derivative values at finitely many points. Those values can also be kept exactly at every approximating stage, provided they belong to the null solution being approximated.

Corollary 2.4 (compatible finite constraints). Keep the homogeneous or degenerate scope of Theorem 2.1. For and any continuous complex-linear map , there are polynomial null solutions with for every . This includes any finite set of derivative values at finitely many points.

Proof. Let be the polynomial null space. Theorem 2.1 gives converging to . Choose a basis of . Finite elimination on the independent columns supplies coordinate rows such that the matrix is invertible. For , its basis coordinates are , a continuous function of . If converges to in , pass to the limit in to get that same identity for . Hence is closed.