arXiv subjects

Valery Asiryan

Publications and source records attributed to Valery Asiryan.

7 recordsLinked to original sources

arXiv2026

Computational Evidence Against Quadratic-Cubic Factorization for the Second Cuboid Quintic

Let $Q_{p,q}(t)\in\mathbb{Z}[t]$ be Sharipov's even monic degree-$10$ second cuboid polynomial depending on coprime integers $p\neq q>0$. Writing $Q_{p,q}(t)$ as a quintic in $t^{2}$ produces an associated monic quintic polynomial. After the weighted normalization $r=p/q$ and $s=r^{2}$ we obtain a one-parameter family $P_s(x)\in\mathbb{Q}[x]$ such that \[ Q_{p,q}(t)=q^{20}\,P_s\!\left(\frac{t^{2}}{q^{4}}\right)\qquad\text{with}\qquad s=\left(\frac{p}{q}\right)^{2}. \] Assuming a quadratic divisor $x^{2}+ax+b$ with $a,b\in\mathbb{Q}$, we reduce divisibility of $P_s(x)$ to the vanishing of an explicit remainder \[ R(x)=R_{1}(s,a,b)\,x+R_{0}(s,a,b). \] A key structural observation is that $R_1$ and $R_0$ are quadratic in $b$ and that, on the equation $R_1=0$, the second condition becomes linear in $b$. This yields a one-direction elimination to a plane obstruction curve $F(s,a)=0$ with $F\in\mathbb{Z}[s,a]$, without any lifting-back issues: when the linear coefficient is nonzero, the parameter $b$ is forced to be the rational value $b=C/L$. We isolate the degenerate locus $L=C=0$ and show it produces only $s=\pm 1$ (hence only $s=1$ in the cuboid domain $s>0$). Let $\overline{C}\subset\mathbb{P}^{2}$ be the projective closure of $F(s,a)=0$. Using Magma we perform a height-bounded search for rational points on $\overline{C}$. With bound $H=10^{9}$, the search returns $8$ rational points, whose affine part has $s\in\{-1,0,1\}$. In particular, no affine rational point with $s>0$ and $s\neq 1$ is found up to this bound. This provides strong computational evidence that for rational $s>0$, $s\neq 1$, the quintic $P_s(x)$ admits no quadratic factor over $\mathbb{Q}$ (equivalently, no $2+3$ (quadratic-cubic) factorization over $\mathbb{Q}$), and yields a conditional exclusion assuming completeness of the rational-point enumeration on $\overline{C}$.

math.GM↗

arXiv2026

Non-Existence of Quintic Factorization for the Second Cuboid Polynomial $Q_{p,q}(t)$

We consider the even monic degree-$10$ second cuboid polynomial $Q_{p,q}(t)\in\mathbb{Z}[t]$ depending on coprime integers $p\neq q>0$. We exclude the existence of a splitting of type $5+5$ over $\mathbb{Q}$, i.e., a factorization of $Q_{p,q}(t)$ into two irreducible quintic polynomials. Since $Q_{p,q}(t)$ is even and satisfies $Q_{p,q}(0)\neq 0$, any such $5+5$ splitting is necessarily symmetric, meaning that it can be written in the normal form $Q_{p,q}(t)=R_{p,q}(t)\cdot (-R_{p,q}(-t))$. After a weighted normalization reducing to a one-parameter polynomial $Q_r(u)$ with $r=p/q\in\mathbb{Q}_{>0}$, coefficient comparison and elimination via resultants show that a $5+5$ splitting forces the existence of a rational point on an explicitly defined plane curve $F(r,a)=0$. Passing to the quotient parameters $a=r y$ and $s=r^2$ yields an affine curve $f(s,y)=0$ such that, for each fixed $s>0$, the polynomial $f(s,\cdot)$ is of degree $16$. We compute and factor the discriminant $\mathrm{Disc}_y(f)$ and then use Sturm root counts to certify that $f(s,\cdot)$ has no real roots for every rational $s>0$ with $s\neq 1$. Hence $f(s,y)=0$ admits no rational solutions with $s>0$, $s\neq 1$, and consequently no quintic $5+5$ factorization occurs for $Q_{p,q}(t)$ when $p\neq q$.

math.GM↗

arXiv2026

Non-Existence of Linear-Quartic Factorization for the Second Cuboid Quintic

Let $Q_{p,q}(t)\in\mathbb{Z}[t]$ be Sharipov's even monic degree-$10$ second cuboid polynomial depending on coprime integers $p\neq q>0$. Writing $Q_{p,q}(t)$ as a quintic in $t^{2}$ produces an associated monic quintic polynomial. After the weighted normalization $r=p/q$ and $s=r^{2}$ we obtain a one-parameter family $P_s(x)\in\mathbb{Q}[x]$ such that \[ Q_{p,q}(t)=q^{20}\,P_s\!\left(\frac{t^{2}}{q^{4}}\right)\qquad\text{with}\qquad s=\left(\frac{p}{q}\right)^{2}. \] We show that for every rational $s>0$ with $s\neq 1$ the equation $P_s(x)=0$ has no rational solutions. Equivalently, $P_s$ admits no $1+4$ factorization over $\mathbb{Q}$. The proof uses an explicit quotient by the inversion involution $(s,y)\mapsto(1/s,1/y)$ and reduces the rational-root problem for $P_s$ to rational points on the fixed genus-$2$ hyperelliptic curve \[ C:\quad w^2=t^5+21t^4+26t^3+10t^2+5t+1=(t+1)(t^4+20t^3+6t^2+4t+1). \] Using Magma and Chabauty's method on the Jacobian of $C$, we compute $C(\mathbb{Q})$ exactly and conclude that the only parameter value producing a rational root is the excluded case $s=1$ (equivalently $p=q$). As a consequence, for coprime $p\neq q>0$ the polynomial $Q_{p,q}(t)$ has no rational roots (hence no linear factor over $\mathbb{Q}$, and in particular no linear factor over $\mathbb{Z}$).

math.GM↗

arXiv2025

Genus-One Fibrations and the Jacobian of Linear Slices in the Quintic Equal-Sum Problem

We study the Diophantine equation $a^5+b^5=c^5+d^5$ under the linear slicing constraint $(c+d)-(a+b)=h$. We first prove the necessary congruence $30\mid h$. After symmetrization, the associated discriminant equation defines, for each fixed nonzero slice parameter $h$, a genus-one curve over $\mathbb{Q}(S)$; to study Mordell-Weil rank, one must pass to its Jacobian fibration $E_h/\mathbb{Q}(S)$. We show that $E_h$ carries a global rational $2$-torsion section and never has full rational $2$-torsion over $\mathbb{Q}(S)$. We also prove that no nonsingular rational specialization acquires additional rational $2$-torsion: by homogeneity, the relevant square condition reduces to rational points on a universal genus-two hyperelliptic curve, whose rational points are determined via a verified Magma computation using a rank-$0$ bound and the Chabauty-Coleman method. We further show that, after the normalization $x=S/h$, the Jacobian fibrations for all $h\neq 0$ become isomorphic over a rational function field. For the representative slice $h=30$, we compute the classical invariants of the associated binary quartic, obtain an explicit Weierstrass model, and apply the Gusi\'c-Tadi\'c injectivity criterion together with verified specialized-rank computations to prove the uniform bound $\mathrm{rank}\,E_h(\mathbb{Q}(S))\le 1$ for all $h\neq 0$. We then construct an explicit rational section on the universal Jacobian model and, specializing at $S=12$ on the slice $h=30$, show via injective specialization that this section has infinite order. Consequently, $\mathrm{rank}\,E_h(\mathbb{Q}(S))=1$ for every $h\in\mathbb{Q}^\times$. We conclude by recording the additional integrality, parity, and size conditions required to recover integer solutions from the genus-one/Jacobian framework.

math.NT↗

arXiv2025

The Linear Slicing Method for Equal Sums of Like Powers: Modular and Geometric Constraints

We study the Diophantine equation $a^k + b^k = c^k + d^k$ with integer variables and exponent $k>1$, under the linear constraint $(c+d) - (a+b) = h$. We analyze the geometry and arithmetic of these linear slices. On the central slice $h=0$, we prove strictly convex uniqueness: distinct unordered pairs with the same sum yield distinct power sums. For shifted slices $h\neq 0$, we establish a Modular Divisibility Obstruction (MDO): any solution requires $h$ to be divisible by a specific squarefree modulus $M_k = \prod_{p-1 \mid k-1} p$. This condition creates a strong divisibility filter; for example, if $k=13$, the obstruction eliminates $99.96\%$ of all possible shifts. We combine this arithmetic constraint with a geometric exclusion zone principle and a global overlap bound, showing that the slice size must satisfy $\min\{S, S+h\} \gg |h|$. Finally, we prove an asymptotic dominance bound $k \le \max\{S, S+h\} \log 2$, implying that for any fixed slice, solutions cannot exist for sufficiently large $k$.

math.NT↗

arXiv2025

Irreducibility of the Cuboid Polynomial $P_{a,u}(t)$ via a Rank-Zero Elliptic Curve

In this paper we study the even monic degree-8 cuboid polynomial $P_{a,u}(t)$ introduced by R.A. Sharipov in the first-cuboid specialization of his cuboid equations. For nonzero integers $a,u$ with $u^2\neq a^2$ we prove that $P_{a,u}(t)$ is irreducible in $\mathbb{Z}[t]$ (equivalently, in $\mathbb{Q}[t]$), thus confirming Sharipov's irreducibility conjecture in this two-parameter case. Over $K=\mathbb{Q}(\sqrt2)$ we have a factorization $P_{a,u}(t)=H_-(t)H_+(t)$ into two conjugate quartics. We show that any further factorization of $H_\pm$ would force the discriminant of a certain quadratic in $S=t^2$ to be a square in $K$, which in turn implies (via $\tau=(au/\Delta)^2$) the existence of a rational point $(y,v)\in\mathcal{C}(\mathbb{Q})$ on the genus-one quartic $\mathcal{C}:\ v^2=16y^4+136y^2+1$ with $y^2=\tau$. We give an explicit isomorphism $\overline{\mathcal{C}}\simeq E$ with the elliptic curve $E:\ Y^2=X(X-8)(X-9)$, whose Mordell-Weil group has rank $0$ and conductor $48$. Enumerating $E(\mathbb{Q})$ and tracing back to $\mathcal{C}(\mathbb{Q})$ rules out the only possible values $\tau\in\{0,\tfrac14\}$, and hence excludes any factorization in $K[t]$. A quadratic Galois descent then yields the irreducibility of $P_{a,u}(t)$ over $\mathbb{Q}$ and $\mathbb{Z}$.

math.GM↗

arXiv2025

On the Irreducibility of the Cuboid Polynomial $P_{a,u}(t)$

In this paper we consider the even monic degree-8 cuboid polynomial $P_{a,u}(t)$ with coprime integers $a\neq u>0$. We prove irreducibility over $\mathbb{Z}$ by excluding all degree-8 splittings. First, any putative $4{+}4$ factorization is shown to force a specific Diophantine constraint that has no integer solutions, via a short $2$- and $3$-adic analysis. Second, we exclude every $2{+}6$ factorization using an exact divisor criterion together with a discriminant obstruction. Finally, after ruling out $2{+}6$, the patterns $2{+}2{+}4$, $2{+}2{+}2{+}2$, and $3{+}3{+}2$ regroup trivially to $2{+}6$ and are therefore impossible. Consequently, $P_{a,u}(t)$ admits no nontrivial factorization in $\mathbb{Z}[t]$.

math.GM↗