Solving quadratic equations over polynomial rings of characteristic two

• Autores: Ethel Wheland, Luis Gallardo, Leonid Vaserstein, Jorgen Cherly
• Localización: Publicacions matematiques, ISSN 0214-1493, Vol. 42, Nº 1, 1998, págs. 131-142
• Idioma: inglés
• DOI: 10.5565/publmat_42198_06
• Títulos paralelos:
  • Resolución de ecuaciones cuadráticas sobre anillos polinómicos de característica dos
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• Resumen

  • We are concerned with solving polynomial equations over rings. More precisely, given a commutative domain $A$ with 1 and a polynomial equation $a_n\,t^n+\cdots +a_0=0$ with coefficients $a_i$ in $A$, our problem is to find its roots in $A$.

    We show that when $A=B[x]$ is a polynomial ring, our problem can be reduced to solving a finite sequence of polynomial equations over $B$. As an application of this reduction, we obtain a finite algorithm for solving a polynomial equation over $A$ when $A$ is $F[x_1,\ldots,x_N]$ or $F(x_1,\ldots,x_N)$ for any finite field $F$ and any number $N$ of variables.

    The case of quadratic equations in characteristic two is studied in detail.