# Polynomial tests and a convergent local Fourier quotient Written by GPT-6.1 Sol (OpenAI), Ultra, October 2026. Reader orientation: CC0-1.0. ## How polynomial tests become an analytic quotient The question starts with a compact distribution that gives zero on every polynomial solution of a constant-coefficient equation. Each polynomial moment is one Taylor coefficient of its Fourier transform. A quotient of formal series records all those moment tests, including every multiplicity. The central work is showing that this formal quotient actually converges near zero. Follow the proof in five steps: 1. **Keep the transpose sign.** LP1–LP2 turn moments into coefficients and polynomial-kernel annihilation into formal divisibility. Complex-linear pairings give the divisor `P(-z)`, with every odd-order sign retained. The algebraic functional used here has no analytic bound yet. 2. **Isolate the roots near zero.** LP3–LP4 choose a regular coordinate direction and one fixed contour. Contour power sums construct the entire inner-root polynomial, including repeated roots and collisions. Individual root branches need not be analytic. 3. **Construct a finite remainder.** LP5 uses the Cauchy integral to write the analytic numerator as the inner factor times an analytic function plus a polynomial remainder. Analyticity alone does not make that remainder vanish. 4. **Use formal divisibility at every parameter order.** LP6 compares coefficients modulo successive powers of the parameter ideal. The only possible low-degree remainder is zero. Dividing by the retained analytic unit gives the actual quotient germ. 5. **Read the convergence bound locally.** LP7 bounds the quotient on a smaller cylinder, then bounds every Taylor coefficient. Geometric series give absolute locally uniform convergence on smaller polydisks. These are bounds for the actual compact moments, rather than for an arbitrary algebraic functional. Example 1 explains why a repeated root needs several moments. Example 2 follows colliding roots without choosing square-root branches. Example 3 keeps a factor that is a unit near zero but has a zero elsewhere: the local quotient has a genuine global pole. Exercise 6 then gives an unrestricted algebraic functional whose formal quotient diverges, identifying the analytic hypothesis that prevents this failure for compact distributions. ## Where this enters the course Choosing polynomial and exponential approximants uses precisely this local polynomial-annihilator entry. That lesson proves its separate local-to-entire continuation under the hypothesis that every irreducible factor vanishes at zero. Compact Fourier division and multiplicity-sensitive annihilators retains the stronger all-center exponential-polynomial annihilator, entire quotient and compact inverse alternative. A local germ alone supplies neither that stronger annihilation nor an entire quotient. The full LP0–LP10 proof, all three examples, three figures and their captions, and all six solutions follow. The written lower entries, compact singular-support hull and broader course obligations retain their stated scope. Exact original files, native figures, reproduction and terms remain available alongside this reader. --- ## Complete formal proof: Polynomial tests and a convergent local Fourier quotient Written by GPT-6.1 Sol (OpenAI), Ultra, October 2026. Original exposition, exercises and reproducible diagrams: CC0-1.0. Classical polynomial duality and local analytic division are explained in full at the stated entries; no novelty of their theorems is claimed. Self-checked by the writing AI. We prove the first Fourier entry of AN02-L011. A purely formal quotient initially records infinitely many polynomial moment conditions. That quotient must then be shown to converge. The proof isolates the roots near the origin, constructs a finite analytic remainder, and proves that formal divisibility forces this remainder to vanish. It does not assume a general analytic preparation or division theorem. ### LP0. Statement and actual entries Let \(n\geq1\), let \(P\in\mathbb C[z_1,\ldots,z_n]\) be nonzero, and put \(D_j=-i\partial_{x_j}\). Distribution pairings are complex linear. For \(\mu\in\mathcal E'(\mathbb R^n)\), its Fourier–Laplace transform is \[ F(z)=\mu(e^{-ix\cdot z}),\qquad Q(z)=P(-z). \tag{LP1} \] **Theorem LP0.** The following are equivalent: 1. \(\mu(h)=0\) for every complex polynomial \(h(x)\) satisfying \(P(D)h=0\). 2. The Taylor series of \(F\) at zero is divisible by \(Q\) in the formal power-series ring. 3. There is a holomorphic function \(G\) on a neighborhood of zero with \(F=QG\) there. The quotient germ in assertion 3 is unique. All polynomial factors and all their multiplicities are retained. If \(P(0)\ne0\), all three assertions hold for every compact \(\mu\): the polynomial kernel is zero and the divisor is a local unit. A nonzero scalar is included. For \(n=0\), the polynomial and test spaces are \(\mathbb C\), the nonzero operator is scalar multiplication, and the same statement is immediate. The zero distribution has the zero quotient; the zero polynomial is excluded. The compact transform, its moments and complex-linear transpose conventions are supplied in the complete compact Fourier proof CF042. In fact the analytic argument below applies to any convergent germ \(F\); compact support is used to provide that analytic germ and its actual moments. The remaining entries are finite polynomial arithmetic, the fundamental theorem of algebra, the one-variable Cauchy formula, holomorphic parameter integration and convergent Taylor expansions. Exact programme providers are *Primitive factors and moving polynomial roots* and *Cauchy bounds, root counts and analytic extensions*. Their declared lower scope is retained. The new argument proves the required local factor and division directly, rather than importing a general Weierstrass theorem. It asserts no additional Sobolev, weak, elliptic or global singular-support result. ### LP1. Every polynomial moment is a formal coefficient Write \(\alpha!=\prod_j\alpha_j!\), \(|\alpha|=\sum_j\alpha_j\), and \(x^\alpha=\prod_jx_j^{\alpha_j}\). Compact distributions act on smooth functions by inserting any compact cutoff equal to one near their support; the result is independent of the cutoff. Differentiating the compact Fourier transform gives \[ \partial_z^\alpha F(0)=(-i)^{|\alpha|}\mu(x^\alpha),\qquad F(z)=\sum_\alpha\frac{(-i)^{|\alpha|}\mu(x^\alpha)}{\alpha!}z^\alpha. \tag{LP2} \] Here the displayed series is convergent because \(F\) is entire. First use the same coefficient rule without assuming convergence. Let \(V=\mathbb C[x_1,\ldots,x_n]\), and let \(V^*\) be its algebraic complex-linear dual. A polynomial has finitely many monomials, so every array of monomial values defines a functional. Thus \[ \begin{split} \mathfrak F:V^*&\longrightarrow\mathbb C[[z_1,\ldots,z_n]],\\ \ell&\longmapsto\sum_\alpha \frac{(-i)^{|\alpha|}\ell(x^\alpha)}{\alpha!}z^\alpha \end{split} \tag{LP3} \] is a linear bijection. Its inverse sends coefficients \(c_\alpha\) to \(\ell(x^\alpha)=i^{|\alpha|}\alpha!c_\alpha\). No boundedness, distribution realization or convergence is asserted for an arbitrary \(\ell\). Let \(T=P(D):V\to V\). For \(P(z)=\sum_\beta p_\beta z^\beta\), its algebraic transpose is \((T^*\ell)(h)=\ell(P(D)h)\). The coefficient of \(z^\alpha\) in \(\mathfrak F(T^*\ell)\) is \[ \begin{split} &\frac{(-i)^{|\alpha|}}{\alpha!} \sum_{\beta\leq\alpha}p_\beta(-i)^{|\beta|} \frac{\alpha!}{(\alpha-\beta)!}\ell(x^{\alpha-\beta})\\ &=\sum_{\beta\leq\alpha}p_\beta(-1)^{|\beta|} \frac{(-i)^{|\alpha-\beta|}\ell(x^{\alpha-\beta})} {(\alpha-\beta)!}. \end{split} \tag{LP4} \] This is precisely the coefficient of \(Q\mathfrak F(\ell)\). Consequently \[ \mathfrak F(T^*\ell)=P(-z)\mathfrak F(\ell)=Q\mathfrak F(\ell). \tag{LP5} \] The two factors \((-i)^{|\beta|}\) in LP4 explain the minus sign in the divisor. Replacing \(P(-z)\) by \(P(z)\) would change the odd-order terms. ### LP2. Annihilating the kernel means formal divisibility We prove the exact algebraic statement used here. For any linear map \(T:V\to V\), \[ \{\ell\in V^*: \ell|_{\ker T}=0\}=\operatorname{im}T^*. \tag{LP6} \] If \(\ell=T^*a\), its value on the kernel is zero. Conversely, if \(\ell\) vanishes on the kernel, define \(a_0\) on \(\operatorname{im}T\) by \(a_0(Th)=\ell(h)\). This is well defined: two preimages differ by a kernel vector. Extend it to \(V\) as follows. Enumerate the countable monomial basis \(b_1,b_2,\ldots\), and put \(W_j=\operatorname{im}T+\operatorname{span}(b_1,\ldots,b_j)\). At step j, if \(b_j\) lies in \(W_{j-1}\), its value is already fixed. Otherwise assign it value zero and extend linearly to the direct sum with \(\mathbb Cb_j\). The compatible functionals on \(W_j\) define a functional a on their union \(V\). Then \(T^*a=\ell\). This construction is algebraic; it is not a continuous Hahn–Banach extension and supplies no convergence bound. Apply LP6 to \(\ell=\mu|_V\), and use the bijection LP3 and the exact transpose LP5. It gives \[ \mu|_{\ker P(D)}=0 \quad\Longleftrightarrow\quad \widehat F=Q\widehat G\text{ for some }\widehat G\in\mathbb C[[z]], \tag{LP7} \] where \(\widehat F\) is the Taylor series of the actual F. This proves assertions 1 and 2 equivalent. Formal quotient uniqueness follows because this series ring has no zero divisors: lowest nonzero homogeneous parts of a product are the product of two nonzero polynomials and cannot vanish. Formal existence alone still does not prove assertion 3. ### LP3. One regular direction and a stable root circle We now prove: for a nonzero polynomial Q and a holomorphic germ F, formal divisibility \(\widehat F=Q\widehat G\) implies a convergent quotient germ. If \(Q(0)\ne0\), ordinary holomorphic reciprocal gives the quotient. Suppose \(Q(0)=0\), and let m be the total degree of Q. Choose a real vector \(\theta\) with its highest homogeneous part \(Q_m(\theta)\ne0\). Such a vector exists: a complex polynomial vanishing on all real vectors is the zero polynomial, by applying the one-variable coefficient conclusion successively to every coordinate. Extend \(\theta\) to a real linear coordinate basis and write \[ z=A(t,w),\qquad w\in\mathbb C^{n-1},\qquad Q(A(t,w))=c t^m+\sum_{j0 small enough that F is holomorphic on a neighborhood of the closed t-disk and zero parameter, and that the only zero of \(Q(t,0)\) in \(|t|\leq R\) is zero. It has multiplicity s. The circle has no zeros, so on a sufficiently small connected parameter polydisk W, \[ |Q(\zeta,w)|\geq\delta_Q>0 \quad(|\zeta|=R,\ w\in\overline W). \tag{LP10} \] F is also holomorphic on a neighborhood of the corresponding closed cylinder. Such W can be made smaller whenever needed below. For k≥0 define the contour moments \[ p_k(w)=\frac1{2\pi i}\int_{|\zeta|=R} \zeta^k\frac{\partial_tQ(\zeta,w)}{Q(\zeta,w)}\,d\zeta. \tag{LP11} \] These are holomorphic in w by uniform parameter differentiation on the fixed compact contour. For each fixed w, the fundamental theorem of algebra factors the actual degree-m polynomial with its multiplicities. Its logarithmic derivative is the sum of \((\zeta-\alpha)^{-1}\) over those roots, with repetitions. Cauchy's formula, or the geometric-series circle integral for an inside or outside root, makes LP11 the sum of \(\alpha^k\) over the roots strictly inside the circle. In particular \(p_0\) is their number with multiplicity. It is an integer and is continuous on the connected W, so equals its value s at zero. This proves the needed persistent root count without choosing holomorphic individual roots or assuming a generic simple-root set. ### LP4. The complete local monic factor Define \(e_0=1\) and, for \(1\leq j\leq s\), set \[ j e_j(w)=\sum_{k=1}^j(-1)^{k-1}e_{j-k}(w)p_k(w),\qquad q(t,w)=\sum_{j=0}^s(-1)^j e_j(w)t^{s-j}. \tag{LP12} \] Every coefficient is holomorphic. To verify the formula rather than assume a root-coefficient theorem, fix the inside roots \(\alpha_1,\ldots,\alpha_s\), with multiplicity. The finite product \(E(v)=\prod_{a=1}^s(1+\alpha_a v)\) has elementary symmetric coefficients. As a formal series at v=0, \(E'(v)/E(v)=\sum_a\alpha_a/(1+\alpha_a v)=\sum_{k\geq1}(-1)^{k-1}p_kv^{k-1}\). Comparing coefficients after multiplying by E gives exactly the recursion in LP12. Its successive divisions by the positive integers j determine those coefficients uniquely. Therefore \[ q(t,w)=\prod_{a=1}^s(t-\alpha_a),\qquad q(t,0)=t^s, \tag{LP13} \] even when roots collide. In particular every coefficient below \(t^s\) vanishes at w=0. The polynomial q includes all inner-root multiplicities and is nonzero on the boundary circle. Long division in t by this monic polynomial gives Q=qb+r, with b and r polynomial in t and holomorphic in w, and \(\deg_t r0\) there. Let \[ M=\sup_{|\zeta|=R,\,w\in\overline{W'}} |F(\zeta,w)/q(\zeta,w)|<\infty. \tag{LP22} \] The length of the contour is \(2\pi R\), and \(|\zeta-t|\geq R-r\). LP15 and LP21 give the explicit bound \[ |H(t,w)|\leq\frac{RM}{R-r},\qquad |G(t,w)|\leq\frac{RM}{\beta(R-r)} \quad(|t|\leq r,\ w\in\overline{W'}). \tag{LP23} \] If \(W'\) has positive coordinate radii \(\rho_1,\ldots,\rho_{n-1}\), the successive one-variable Cauchy formulas on its product contours and the t-contour bound the Taylor coefficient \(g_{a,\gamma}\) of G by \[ |g_{a,\gamma}|\leq \frac{RM}{\beta(R-r)}r^{-a}\prod_j\rho_j^{-\gamma_j}. \tag{LP24} \] Using slightly smaller radii if the boundary is only specified as a closed cylinder gives the same conclusion. Products of geometric series now give absolute locally uniform convergence on every strictly smaller polydisk. This is a quantitative convergence mechanism for the actual formal quotient, not an assumption about the unrestricted algebraic extension a in LP2. The theorem gives a germ near zero. It does not imply that the quotient is entire, that an arbitrary algebraic functional is a compact distribution, or that the divisor's factors all meet zero. The complete CF042 theorem supplies the distinct entire quotient and compact inverse equivalence when the stronger exponential-polynomial annihilation holds. L011's continuation argument supplies the separate implication from a local germ to an entire quotient when every irreducible factor vanishes at zero. Those hypotheses must remain visible. ![The exact polynomial moment, transpose and analytic remainder maps](../reproduce/L123/figures/moments-to-local-quotient.png) **Figure 1.** This schematic displays the actual maps of LP2–LP7 and LP15–LP21. Every moment coefficient retains its factorial and complex phase; the divisor is P(-z). The middle algebraic functional need not be a distribution. Analyticity of F and vanishing of the finite remainder, proved by parameter-order induction, produce the convergent germ G=H/b. The boxes are a map schematic, not a finite truncation of the moment conditions. ### LP8. Three worked examples **Example 1: a repeated zero in one variable.** For \(P(z)=z^m\), m≥1, polynomial solutions are exactly those of degree below m: a polynomial of degree at least m has a nonzero mth derivative. Since \(Q(z)=(-1)^mz^m\), Theorem LP0 says \[ \mu(x^j)=0\ (0\leq j