The test ideal of the Frobenius form

In this article, we explore the Frobenius form f, a class of homogeneous polynomials within the polynomial ring k[x1,…,xn] over an algebraically closed field k of positive characteristic p. We investigate the test ideal of rings defined by the quotient associated with a Frobenius form. Furthermore, we show that the test ideal of the quotient ring R=k[x1,…,xn]/(f) is the image of the ideal (x1,…,xr)[pe−1] under a canonical projection from k[x1,…,xn] to R, where e is an integer and f is rank r Frobenius form of degree pe+1, where e>0.