Let $\mathbb C[t_1,\dots, t_d]$ be the multivariate polynomial ring with a certain monomial order $\leq$. I will denote monomials by $x^\alpha$, where $\alpha \in \mathbb N^d$. For $f = \sum_\alpha a_\alpha x^\alpha \in \mathbb C[t_1,\dots, t_d]$, the notation $LT(f) = a_\alpha x^\alpha, a_\alpha \neq 0, $ means the leading term of $f$ given by the term of whose monomial is maximal with respect to $\leq$. We write $LM(f)= x^\alpha$ to denote the leading monomial of $f$ obtained by dividing $LT(f)$ by its coefficient.
Let $I = \langle f_1,\dots, f_n \rangle,$ where $f_i \in \mathbb C[t_1,\dots,t_d]$. Define $\tilde f_i$ to be an "upper triangular deformation" of $f_i$ of the form $\tilde f_i = f_i + g_i$, where $g_i\in \mathbb C[t_1,\dots, t_d]$ satisfies $LM(f_i)\geq LM(g_i)$. Set $\tilde I = \langle \tilde f_1,\dots, \tilde f_n\rangle$.
Is it true that $\dim \mathbb C[t_1,\dots, t_d]/ I \leq \dim \mathbb C[t_1,\dots, t_d]/ \tilde I$ as $\mathbb C$-vector spaces?
I believe that the leading terms of elements of $I$ and $\tilde I$ are related as follows $$LT(\tilde I) = \{ LT(g): g \in \tilde I \} \subseteq LT(I) = \{LT(f) : f\in I \}.$$ Indeed, any $ h \in \tilde I$ writes as $h = \sum u_i \tilde f_i = \sum u_i f_i + \sum u_ig_i$. By the condition $LM(f_i)\geq LM(g_i)$, it seems that $LT(h) = LT(\sum u_i f_i)$. Since $\sum u_if_i \in I$, then $LT(h) \in LT(I)$. Hence, $LT(\tilde I) \subseteq LT( I) $ and we have the containment of ideals $\langle LT(\tilde I) \rangle \subseteq \langle LT(I) \rangle$.
Now I want to use Groebner basis to finish the problem. Let $G$ be a Groebner basis of $I$. Then, we know that any $f\in \mathbb C[t_1,\dots, t_d]$ writes as $f = q+r$ for some $q\in I$ and unique $r$, where $r\in Span(x^\alpha: x^\alpha \not \in \langle LT(I) \rangle)$. Thus, a basis of $\mathbb C[t_1,\dots, t_d]/I$ is in bijection with $\{x^\alpha: x^\alpha \not \in \langle LT(I) \rangle\}$. Similarly, a basis of $\mathbb C[t_1,\dots, t_d]/\tilde I$ is in bijection with $\{x^\alpha: x^\alpha \not \in \langle LT(\tilde I) \rangle \}$.
From $\langle LT(\tilde I) \rangle \subseteq \langle LT(I) \rangle,$ we get $\{x^\alpha: x^\alpha \not \in \langle LT( I) \rangle\} \subseteq \{ x^\alpha: x^\alpha \not \in \langle LT(\tilde I) \rangle \}$.
Thus, $\dim \mathbb C[t_1,\dots, t_d]/I \leq \dim \mathbb C[t_1,\dots, t_d]/\tilde I$
Is this reasoning correct? I have not much experience with Groebner basis and I would appreciate some help