Fix positive integers $m, n, d$. In what follows, the height of an algebraic number will mean the absolute multiplicative height.

Let $V \subset \bar{\mathbb{Q}}^n$ be an affine algebraic variety such that $$ V = \{ \mathbf{x} \in \bar{\mathbb{Q}}^n \mid f_1(\mathbf{x}) = \dotsb = f_m(\mathbf{x}) = 0 \} $$ for some polynomials $f_1, \dotsc, f_m$ each of degree at most $d$, with coefficients in $\bar{\mathbb{Q}}$ of height at most $H$. We do not require that the ideal $(f_1, \dotsc, f_m)$ should be radical.

Let $\mathop{\mathrm{Sing}} V$ denote the set of singular points of $V$.

Do there exist constants $m', d', a, b$ depending only on $m, n, d$ such that: $$ \mathop{\mathrm{Sing}} V = \{ \mathbf{x} \in \bar{\mathbb{Q}}^n \mid g_1(\mathbf{x}) = \dotsb = g_{m'}(\mathbf{x}) = 0 \} $$ for some polynomials $g_1, \dotsc, g_{m'}$ each of degree at most $d'$, with coefficients in $\bar{\mathbb{Q}}$ of height at most $aH^b$?

Note that I don't care how $m', d', a, b$ depend on $m, n, d$.

If the ideal $I = (f_1, \dotsc, f_m)$ is radical, then the answer is easily yes because then $\mathop{\mathrm{Sing}} V$ is defined by a condition on the Jacobians of $f_1, \dotsc, f_m$. In the general case, I suspect that one might be able to use Gröbner bases to show that the radical of $I$ is generated by a set of polynomials of constant-bounded degrees and polynomially-bounded heights. But I don't know much about Gröbner bases and I haven't found anything which discusses height bounds for Gröbner bases.

I would appreciate comments from experts on whether this will work, or has been done before. Other suggestions are also welcome.

This came up when trying to prove a result about the existence of real points of bounded height on an affine variety defined over $\bar{\mathbb{Q}} \cap \mathbb{R}$. My proof does not work if all the real points of the variety are contained in the singular locus, so I would like to use the above to say that in this case we can replace our initial variety by its singular locus while retaining control of heights of the defining equations.


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