Vieta Migration: Root Transformation and Group Structure under Coefficient Permutation for Polynomial Equations

Vieta Migration: Root Transformation and Group Structure under Coefficient Permutation for Polynomial Equations Classical Vieta theory relates the coefficients of a fixed polynomial equation to its roots. This paper studies the complementary question: when coefficients are permuted, how do the roots transform, and what group-theoretic structure governs this transformation? We develop a systematic framework of Vieta migration, defined as the algebraic correspondence\[\Phi_\sigma = V^{-1} \circ p_\sigma \circ V,\]where \(V\) is the Vieta map from roots to coefficients and \(p_\sigma\) is a coefficient permutation. This correspondence transports root data from one equation to another within a finite coefficient orbit. The composition law\[\Phi_{\pi\circ\rho} = \Phi_\rho \circ \Phi_\pi\]makes Vieta migration a genuine group action on the level of algebraic correspondences, compatible with the coefficient-permutation group \(\mathfrak S_n\). The identity \(\Phi_{\mathrm{id}}=\mathrm{id}\) and the inverse relation \(\Phi_{\sigma^{-1}}=\Phi_\sigma^{-1}\) complete the groupoid structure: the equations in the orbit are the objects, the migrations are the arrows, and each object carries its own Galois group \(\Gal(f_a^\sigma/K)\) as its automorphism group. Thus the Galois groups of all equations in a coefficient orbit assemble into a Galois migration groupoid whose objects are Galois groups themselves. On each coefficient-permutation orbit \(\mathcal O(f_a)=\{f_a^\sigma:\sigma\in\mathfrak S_n\}\), familiar polynomial properties—reducibility, rationality of roots, and solvability of Galois groups—are not intrinsic to a single equation but are local data on this orbit. They vary as Boolean-valued functions\[P_f:\mathcal O(f_a)\to\{\mathrm{true},\mathrm{false}\}.\]Explicit quadratic examples and two independent quintic counterexamples demonstrate that these properties can change under coefficient permutation. We further introduce arithmetic closure expansion, where the coefficient set is closed under pairwise addition, subtraction, multiplication, and division by nonzero elements. In characteristic zero this closure contains \(\mathbb Q\), giving every rational-coefficient degree-\(n\) equation a structured Vieta path from any nonzero starting equation. This reveals that the roots of all rational-coefficient equations of the same degree are connected through a global network of Vieta correspondences. The coefficient-generation path, the Vieta fiber, and the induced splitting field data together form a structured root access path between arbitrary rational equations. Thus Vieta migration transforms the study of polynomial equations from the isolated search for roots of a single equation to the structural analysis of how roots transform, how groups act on the entire coefficient-induced equation space, and how all rational-coefficient equations of the same degree are connected through arithmetic closure. The framework naturally extends to \(n\)-variable degree-\(m\) equations: coefficient permutations act on monomial-position coefficients, and the ordinary Vieta migration theorem applies to every generic finite projection fiber. --- Extended Abstract **Induced Coefficient Rearrangement and Groupoid Analysis on One-Variable Equations with Integral Coefficients** This paper develops a comprehensive theory of coefficient rearrangement as an external symmetry operation dual to the classical internal symmetry of roots studied by Galois theory. The central observation is that while classical Galois theory permutes the roots of a fixed equation while keeping coefficients fixed, one may instead permute the coefficients themselves while the powers of the variable remain fixed. This operation is not a permutation of roots—it generally produces a new equation, a new Vieta fiber, and possibly a different splitting field or Galois group. The paper systematically investigates the algebraic structures induced by this rearrangement and establishes a complete framework that unifies Vieta correspondence, coefficient migration, Galois groupoids, arithmetic closure, and a measure-theoretic resolution of a Schanuel-type algebraic-independence problem. The paper is organized in six interconnected parts. **Part I** establishes the foundational migration formulas. For an integral polynomial with coefficient vector \(c=(c_0,\ldots,c_n)\), a permutation \(\pi\in\mathfrak S_{n+1}\) sends the coefficients to new positions. The paper proves the general full-coefficient migration theorem:\[u_k=(-1)^{k+\pi(n)-\pi(n-k)}\frac{s_{n-\pi(n-k)}}{s_{n-\pi(n)}},\qquad 0\le k\le n,\]where \(s_k\) are the elementary symmetric functions of the original roots and \(u_k\) are those of the migrated roots. This formula shows that coefficient permutation changes the Vieta data by a signed ratio of the original symmetric functions. The monic fixed-leading-coefficient specialization yields the simpler form\[t_k=(-1)^{k+n-\sigma(n-k)}s_{n-\sigma(n-k)},\qquad 1\le k\le n.\]Explicit quadratic migration formulas, ratio and product identities, and direct root-migration formulas via Vieta-root solver composition are derived. These results establish that Vieta migration extends the classical Vieta theorem from an internal relation of one equation to an external relation among equations in a coefficient-permutation orbit. **Part II** studies the finite coefficient-permutation orbit as an equation space. Properties such as reducibility, rational splitting, radical solvability, splitting fields, and Galois groups are shown to be local functions on this orbit rather than global invariants of the coefficient multiset. The paper constructs the coefficient-orbit Galois groupoid: objects are the Galois data attached to each permuted equation, and arrows are the Vieta migration correspondences \(\Phi_\sigma=V^{-1}\circ p_\sigma\circ V\), which satisfy identities, inverses, and composition laws inherited from coefficient permutations. This groupoid formulation captures precisely the “complex orbit” phenomenon: the number of roots remains \(n\), while the attached algebraic data may expand, contract, or change type. Single-root migration orbits, Galois-type strata, and setwise stabilizers are analyzed. Quintic examples demonstrate that coefficient permutation can change radical solvability even when restricted to nonzero coefficients, moving a solvable reducible quintic to an irreducible nonsolvable quintic with Galois group \(S_5\). **Part III** enlarges the finite orbit through legal normalization and arithmetic closure. Legal normalization—dividing by any existing nonzero coefficient—creates expansion towers of equations. The finite stabilization theorem proves that the legal expansion state space is bounded by \((n+1)!|S(c)|\), establishing convergence in the sense of state-set stabilization. Vieta paths connect vertices of the expansion graph through algebraic correspondences. Galois towers of expanded equations are defined layerwise: \(L_r(F_c)\) is the compositum of splitting fields of all equations at depth \(\le r\), and \(\mathcal G_r(F_c)=\Gal(L_r(F_c)/K)\). Restriction maps \(\mathrm{res}_{r+1,r}\) form an inverse system of finite Galois groups. Arithmetic closure expansion allows arbitrary finite field operations on coefficients; in characteristic zero, the arithmetic closure contains \(\mathbb Q\), so every rational-coefficient equation of the same degree appears as a structured coefficient target. Real completion gives the completed coefficient state space \(\mathcal C_n(\mathbb R)=\mathbb R^n\), whose unlabelled root range is the entire complex plane. Projective compactification yields circle, web, and disk models of the arithmetic root network's boundary topology. **Part IV** generalizes the entire architecture to multivariable algebraic equations. Replacing the one-variable exponent index with the multi-index monomial set \(I_{n,m}=\{\alpha\in\mathbb N^n:|\alpha|\le m\}\) gives the corresponding theory for \(n\)-variable equations of total degree at most \(m\). The multivariable migration theorem shows that, on ordinary projection fibers, the correspondence restricts to the one-variable Vieta migration formalism over the rational function field of the projection base. The global equation space \(\mathbb P\mathcal C_{n,m}^{(m)}(K)\) and solution-orbit networks are analyzed. The completed real coefficient base is homeomorphic to \(\mathbb RP^{N_{n,m}-1}\setminus\mathbb RP^{N_{n,m-1}-1}\), where \(N_{n,m}=\binom{n+m}{m}\). The universal solution incidence space \(\mathfrak X_{n,m}^{\mathbb R}=\{([c],X):F_c(X)=0\}\) is semialgebraic. A standard research protocol organizes the study of arbitrary multivariable equations through the same three-layer method. **Part V** presents the Schanuel-type application. The paper establishes a projection criterion reducing a two-variable algebraic relation \(P(\pi,e)=0\) to the membership \(\pi\in\overline{\mathbb Q(e)}\), which is further identified with a one-variable Vieta fiber over \(\mathbb Q(e)\). Finite migrated exponential layers are shown to have rational coefficients, while infinite migrated exponentials \(E_\sigma(Z)=\sum_{n=0}^{\infty}Z^n/\sigma(n)!\) are governed by the radius criterion \(R_\sigma=\exp(\liminf g_\sigma(n))\), where \(g_\sigma(n)=\log(\sigma(n)!)/n\). The three-class decomposition partitions infinite-support permutations into \(\mathcal C_1\) (finite radius), \(\mathcal C_2\) (infinite radius with algebraic correlation \(I_\sigma\ne(0)\)), and \(\mathcal C_3\) (infinite radius without correlation). The expansion-tower bridge theorem proves the unconditional equivalence:\[\mathsf S_{\pi,e}\quad\Longleftrightarrow\quad\mathsf M_{\pi}^{\mathrm{alg}},\]where \(\mathsf S_{\pi,e}\) is the classical algebraic-dependence predicate and \(\mathsf M_{\pi}^{\mathrm{alg}}\) asserts the existence of \(\sigma\in\mathfrak S_{\mathbb N}^{\infty}\) and \(t\in\mathbb Q_{>1}\) with \(\pi\in\overline{\mathbb Q(E_\sigma